{"id":696,"date":"2025-07-14T21:10:12","date_gmt":"2025-07-14T21:10:12","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=696"},"modified":"2026-01-15T17:49:12","modified_gmt":"2026-01-15T17:49:12","slug":"module-10-matrices-and-matrix-operations-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-10-matrices-and-matrix-operations-cheat-sheet\/","title":{"raw":"Matrices and Matrix Operations: Cheat Sheet","rendered":"Matrices and Matrix Operations: Cheat Sheet"},"content":{"raw":"<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Essential Concepts<\/h2>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Matrices and Matrix Operations<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Matrix Basics<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">A matrix is a rectangular array of numbers arranged in rows and columns<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Dimensions are written as [latex]m \\times n[\/latex], where [latex]m[\/latex] = number of rows and [latex]n[\/latex] = number of columns<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Elements are the individual numbers in the matrix, often denoted [latex]a_{ij}[\/latex] where [latex]i[\/latex] is the row and [latex]j[\/latex] is the column<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Matrix Addition and Subtraction<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Matrices must have equal dimensions to be added or subtracted<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Add or subtract corresponding entries from each matrix<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Scalar Multiplication<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Multiply every entry in the matrix by the scalar (constant)<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Matrix Multiplication<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Only possible when inner dimensions match: the number of columns in the first matrix must equal the number of rows in the second matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]A[\/latex] is [latex]m \\times n[\/latex] and [latex]B[\/latex] is [latex]n \\times p[\/latex], then [latex]AB[\/latex] is [latex]m \\times p[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">To find entry [latex]c_{ij}[\/latex] of product matrix [latex]C[\/latex]:\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Multiply each entry in row [latex]i[\/latex] of [latex]A[\/latex] by corresponding entry in column [latex]j[\/latex] of [latex]B[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Sum all products: [latex]c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \\cdots + a_{in}b_{nj}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Matrix multiplication is not commutative: [latex]AB \\neq BA[\/latex] in general<\/li>\r\n<\/ul>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Solving Systems with Gaussian Elimination<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Augmented Matrices<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">An augmented matrix represents a system of equations using coefficients and constants<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Written as [latex][A|B][\/latex] where [latex]A[\/latex] contains coefficients and [latex]B[\/latex] contains constants<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Example: [latex]\\begin{cases} 2x + 3y = 5 \\\\ x - y = 1 \\end{cases}[\/latex] becomes [latex]\\left[\\begin{array}{cc|c} 2 &amp; 3 &amp; 5 \\\\ 1 &amp; -1 &amp; 1 \\end{array}\\right][\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Row Operations<\/strong> Three types of row operations can be performed:<\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Multiply a row by a nonzero constant<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Add a multiple of one row to another row<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Interchange (swap) two rows<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Row-Echelon Form<\/strong> A matrix is in row-echelon form when:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">All rows of zeros (if any) are at the bottom<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The first nonzero entry in each row (called a leading entry) is to the right of the leading entry in the row above<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">All entries below a leading entry are zeros<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Gaussian Elimination Process<\/strong><\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Write the system as an augmented matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Use row operations to obtain row-echelon form<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Back-substitute to find solutions, starting from the bottom row<\/li>\r\n<\/ol>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Solving Systems with Inverses<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Identity Matrix<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Denoted [latex]I[\/latex], has 1's on the main diagonal and 0's elsewhere<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Property: [latex]AI = IA = A[\/latex] for any compatible matrix [latex]A[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]2 \\times 2[\/latex] identity: [latex]\\left[\\begin{array}{cc} 1 &amp; 0 \\\\ 0 &amp; 1 \\end{array}\\right][\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Inverse Matrix<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">The inverse of matrix [latex]A[\/latex] is denoted [latex]A^{-1}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Property: [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Not all matrices have inverses; a matrix with an inverse is called invertible<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]\\det(A) = 0[\/latex], the matrix has no inverse<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Finding the Inverse of a [latex]2 \\times 2[\/latex] Matrix<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For [latex]A = \\left[\\begin{array}{cc} a &amp; b \\\\ c &amp; d \\end{array}\\right][\/latex], the inverse is [latex]A^{-1} = \\frac{1}{ad-bc}\\left[\\begin{array}{cc} d &amp; -b \\\\ -c &amp; a \\end{array}\\right][\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Finding the Inverse Using Row Operations<\/strong><\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Form the augmented matrix [latex][A|I][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Use row operations to transform the left side into [latex]I[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The right side becomes [latex]A^{-1}[\/latex]: [latex][A|I] \\rightarrow [I|A^{-1}][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]A[\/latex] cannot be transformed to [latex]I[\/latex], then [latex]A[\/latex] has no inverse<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solving Systems Using Inverses<\/strong><\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Write the system as [latex]AX = B[\/latex], where [latex]A[\/latex] = coefficient matrix, [latex]X[\/latex] = variable matrix, [latex]B[\/latex] = constant matrix<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Multiply both sides by [latex]A^{-1}[\/latex]: [latex]A^{-1}AX = A^{-1}B[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Simplify: [latex]IX = A^{-1}B[\/latex], so [latex]X = A^{-1}B[\/latex]<\/li>\r\n<\/ol>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Solving Systems with Cramer's Rule<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Determinants<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">The determinant of a [latex]2 \\times 2[\/latex] matrix is: [latex]\\det\\left[\\begin{array}{cc} a &amp; b \\\\ c &amp; d \\end{array}\\right] = ad - bc[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">For a [latex]3 \\times 3[\/latex] matrix, augment with the first two columns and use the diagonal method:\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Add the three products of diagonals going from upper-left to lower-right<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Subtract the three products of diagonals going from lower-left to upper-right<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Cramer's Rule<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For the system [latex]AX = B[\/latex]:<\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]D = \\det(A)[\/latex] (determinant of coefficient matrix)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]D_x = \\det[\/latex] of matrix with [latex]x[\/latex]-column replaced by constant column [latex]B[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]D_y = \\det[\/latex] of matrix with [latex]y[\/latex]-column replaced by constant column [latex]B[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Solutions: [latex]x = \\frac{D_x}{D}[\/latex], [latex]y = \\frac{D_y}{D}[\/latex] (provided [latex]D \\neq 0[\/latex])<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<table id=\"eip-id1165137848559\" style=\"width: 99.2161%;\" summary=\"..\"><colgroup> <col \/> <col \/> <\/colgroup>\r\n<tbody>\r\n<tr valign=\"middle\">\r\n<td style=\"width: 37.7404%;\">Identity matrix for a [latex]2\\text{}\\times \\text{}2[\/latex] matrix<\/td>\r\n<td style=\"width: 61.0136%;\">[latex]{I}_{2}=\\left[\\begin{array}{cc}1&amp; 0\\\\ 0&amp; 1\\end{array}\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"middle\">\r\n<td style=\"width: 37.7404%;\">Identity matrix for a [latex]\\text{3}\\text{}\\times \\text{}3[\/latex] matrix<\/td>\r\n<td style=\"width: 61.0136%;\">[latex]{I}_{3}=\\left[\\begin{array}{ccc}1&amp; 0&amp; 0\\\\ 0&amp; 1&amp; 0\\\\ 0&amp; 0&amp; 1\\end{array}\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"middle\">\r\n<td style=\"width: 37.7404%;\">Multiplicative inverse of a [latex]2\\text{}\\times \\text{}2[\/latex] matrix<\/td>\r\n<td style=\"width: 61.0136%;\">[latex]{A}^{-1}=\\frac{1}{ad-bc}\\left[\\begin{array}{cc}d&amp; -b\\\\ -c&amp; a\\end{array}\\right],\\text{ where }ad-bc\\ne 0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2>Glossary<\/h2>\r\n<dl id=\"fs-id1165134073074\" class=\"definition\">\r\n \t<dt id=\"fs-id1165134073079\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">augmented matrix<\/span><\/dt>\r\n<\/dl>\r\n<dl id=\"fs-id1165134279527\" class=\"definition\">\r\n \t<dd id=\"fs-id1165134279532\">a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165133103268\" class=\"definition\">\r\n \t<dt id=\"fs-id1165133103274\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">coefficient matrix<\/span><\/dt>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt id=\"fs-id1165131866861\">\r\n<dl id=\"fs-id1165133103268\" class=\"definition\">\r\n \t<dd id=\"fs-id1165133103274\">a matrix that contains only the coefficients from a system of equations<\/dd>\r\n<\/dl>\r\n<\/dt>\r\n<\/dl>\r\n<dl id=\"fs-id1165134073074\" class=\"definition\">\r\n \t<dt>column<\/dt>\r\n \t<dd id=\"fs-id1165134073079\">a set of numbers aligned vertically in a matrix<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1674058\" class=\"definition\">\r\n \t<dt>Cramer\u2019s Rule<\/dt>\r\n \t<dd id=\"fs-id1674063\">a method for solving systems of equations that have the same number of equations as variables using determinants<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1674068\" class=\"definition\">\r\n \t<dt>determinant<\/dt>\r\n \t<dd id=\"fs-id1674074\">a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135639171\" class=\"definition\">\r\n \t<dt>element<\/dt>\r\n \t<dd id=\"fs-id1165135639177\">a number in a matrix<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134245049\" class=\"definition\">\r\n \t<dt>Gaussian elimination<\/dt>\r\n \t<dd id=\"fs-id1165134245055\">using elementary row operations to obtain a matrix in row-echelon form<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134179616\" class=\"definition\">\r\n \t<dt>identity matrix<\/dt>\r\n \t<dd id=\"fs-id1165134179622\">a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135639180\" class=\"definition\">\r\n \t<dt>matrix<\/dt>\r\n \t<dd id=\"fs-id1165137937501\">a rectangular array of numbers<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134179626\" class=\"definition\">\r\n \t<dt>multiplicative inverse of a matrix<\/dt>\r\n \t<dd id=\"fs-id1165135528440\">a matrix that, when multiplied by the original, equals the identity matrix<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134033250\" class=\"definition\">\r\n \t<dt>row<\/dt>\r\n \t<dd id=\"fs-id1165137937510\">a set of numbers aligned horizontally in a matrix<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134493421\" class=\"definition\">\r\n \t<dt>row-echelon form<\/dt>\r\n \t<dd id=\"fs-id1165134486699\">after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165134486705\" class=\"definition\">\r\n \t<dt>row-equivalent<\/dt>\r\n \t<dd id=\"fs-id1165131866861\">two matrices [latex]A[\/latex] and [latex]B[\/latex] are row-equivalent if one can be obtained from the other by performing basic row operations<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165137456553\" class=\"definition\">\r\n \t<dt>row operations<\/dt>\r\n \t<dd id=\"fs-id1165132961357\">adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-id1165135199312\" class=\"definition\">\r\n \t<dt>scalar multiple<\/dt>\r\n \t<dd id=\"fs-id1165135199316\">an entry of a matrix that has been multiplied by a scalar<\/dd>\r\n<\/dl>","rendered":"<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Essential Concepts<\/h2>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Matrices and Matrix Operations<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Matrix Basics<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">A matrix is a rectangular array of numbers arranged in rows and columns<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Dimensions are written as [latex]m \\times n[\/latex], where [latex]m[\/latex] = number of rows and [latex]n[\/latex] = number of columns<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Elements are the individual numbers in the matrix, often denoted [latex]a_{ij}[\/latex] where [latex]i[\/latex] is the row and [latex]j[\/latex] is the column<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Matrix Addition and Subtraction<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Matrices must have equal dimensions to be added or subtracted<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Add or subtract corresponding entries from each matrix<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Scalar Multiplication<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Multiply every entry in the matrix by the scalar (constant)<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Matrix Multiplication<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Only possible when inner dimensions match: the number of columns in the first matrix must equal the number of rows in the second matrix<\/li>\n<li class=\"whitespace-normal break-words pl-2\">If [latex]A[\/latex] is [latex]m \\times n[\/latex] and [latex]B[\/latex] is [latex]n \\times p[\/latex], then [latex]AB[\/latex] is [latex]m \\times p[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">To find entry [latex]c_{ij}[\/latex] of product matrix [latex]C[\/latex]:\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Multiply each entry in row [latex]i[\/latex] of [latex]A[\/latex] by corresponding entry in column [latex]j[\/latex] of [latex]B[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Sum all products: [latex]c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \\cdots + a_{in}b_{nj}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Matrix multiplication is not commutative: [latex]AB \\neq BA[\/latex] in general<\/li>\n<\/ul>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Solving Systems with Gaussian Elimination<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Augmented Matrices<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">An augmented matrix represents a system of equations using coefficients and constants<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Written as [latex][A|B][\/latex] where [latex]A[\/latex] contains coefficients and [latex]B[\/latex] contains constants<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Example: [latex]\\begin{cases} 2x + 3y = 5 \\\\ x - y = 1 \\end{cases}[\/latex] becomes [latex]\\left[\\begin{array}{cc|c} 2 & 3 & 5 \\\\ 1 & -1 & 1 \\end{array}\\right][\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Row Operations<\/strong> Three types of row operations can be performed:<\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Multiply a row by a nonzero constant<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Add a multiple of one row to another row<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Interchange (swap) two rows<\/li>\n<\/ol>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Row-Echelon Form<\/strong> A matrix is in row-echelon form when:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">All rows of zeros (if any) are at the bottom<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The first nonzero entry in each row (called a leading entry) is to the right of the leading entry in the row above<\/li>\n<li class=\"whitespace-normal break-words pl-2\">All entries below a leading entry are zeros<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Gaussian Elimination Process<\/strong><\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Write the system as an augmented matrix<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Use row operations to obtain row-echelon form<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Back-substitute to find solutions, starting from the bottom row<\/li>\n<\/ol>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Solving Systems with Inverses<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Identity Matrix<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Denoted [latex]I[\/latex], has 1&#8217;s on the main diagonal and 0&#8217;s elsewhere<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Property: [latex]AI = IA = A[\/latex] for any compatible matrix [latex]A[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]2 \\times 2[\/latex] identity: [latex]\\left[\\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array}\\right][\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Inverse Matrix<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">The inverse of matrix [latex]A[\/latex] is denoted [latex]A^{-1}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Property: [latex]AA^{-1} = A^{-1}A = I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Not all matrices have inverses; a matrix with an inverse is called invertible<\/li>\n<li class=\"whitespace-normal break-words pl-2\">If [latex]\\det(A) = 0[\/latex], the matrix has no inverse<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Finding the Inverse of a [latex]2 \\times 2[\/latex] Matrix<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For [latex]A = \\left[\\begin{array}{cc} a & b \\\\ c & d \\end{array}\\right][\/latex], the inverse is [latex]A^{-1} = \\frac{1}{ad-bc}\\left[\\begin{array}{cc} d & -b \\\\ -c & a \\end{array}\\right][\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Finding the Inverse Using Row Operations<\/strong><\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Form the augmented matrix [latex][A|I][\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Use row operations to transform the left side into [latex]I[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The right side becomes [latex]A^{-1}[\/latex]: [latex][A|I] \\rightarrow [I|A^{-1}][\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">If [latex]A[\/latex] cannot be transformed to [latex]I[\/latex], then [latex]A[\/latex] has no inverse<\/li>\n<\/ol>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Solving Systems Using Inverses<\/strong><\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Write the system as [latex]AX = B[\/latex], where [latex]A[\/latex] = coefficient matrix, [latex]X[\/latex] = variable matrix, [latex]B[\/latex] = constant matrix<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Multiply both sides by [latex]A^{-1}[\/latex]: [latex]A^{-1}AX = A^{-1}B[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Simplify: [latex]IX = A^{-1}B[\/latex], so [latex]X = A^{-1}B[\/latex]<\/li>\n<\/ol>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Solving Systems with Cramer&#8217;s Rule<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Determinants<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">The determinant of a [latex]2 \\times 2[\/latex] matrix is: [latex]\\det\\left[\\begin{array}{cc} a & b \\\\ c & d \\end{array}\\right] = ad - bc[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">For a [latex]3 \\times 3[\/latex] matrix, augment with the first two columns and use the diagonal method:\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Add the three products of diagonals going from upper-left to lower-right<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Subtract the three products of diagonals going from lower-left to upper-right<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Cramer&#8217;s Rule<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For the system [latex]AX = B[\/latex]:<\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">[latex]D = \\det(A)[\/latex] (determinant of coefficient matrix)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]D_x = \\det[\/latex] of matrix with [latex]x[\/latex]-column replaced by constant column [latex]B[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]D_y = \\det[\/latex] of matrix with [latex]y[\/latex]-column replaced by constant column [latex]B[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Solutions: [latex]x = \\frac{D_x}{D}[\/latex], [latex]y = \\frac{D_y}{D}[\/latex] (provided [latex]D \\neq 0[\/latex])<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<table id=\"eip-id1165137848559\" style=\"width: 99.2161%;\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/> <\/colgroup>\n<tbody>\n<tr valign=\"middle\">\n<td style=\"width: 37.7404%;\">Identity matrix for a [latex]2\\text{}\\times \\text{}2[\/latex] matrix<\/td>\n<td style=\"width: 61.0136%;\">[latex]{I}_{2}=\\left[\\begin{array}{cc}1& 0\\\\ 0& 1\\end{array}\\right][\/latex]<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td style=\"width: 37.7404%;\">Identity matrix for a [latex]\\text{3}\\text{}\\times \\text{}3[\/latex] matrix<\/td>\n<td style=\"width: 61.0136%;\">[latex]{I}_{3}=\\left[\\begin{array}{ccc}1& 0& 0\\\\ 0& 1& 0\\\\ 0& 0& 1\\end{array}\\right][\/latex]<\/td>\n<\/tr>\n<tr valign=\"middle\">\n<td style=\"width: 37.7404%;\">Multiplicative inverse of a [latex]2\\text{}\\times \\text{}2[\/latex] matrix<\/td>\n<td style=\"width: 61.0136%;\">[latex]{A}^{-1}=\\frac{1}{ad-bc}\\left[\\begin{array}{cc}d& -b\\\\ -c& a\\end{array}\\right],\\text{ where }ad-bc\\ne 0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>Glossary<\/h2>\n<dl id=\"fs-id1165134073074\" class=\"definition\">\n<dt id=\"fs-id1165134073079\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">augmented matrix<\/span><\/dt>\n<\/dl>\n<dl id=\"fs-id1165134279527\" class=\"definition\">\n<dd id=\"fs-id1165134279532\">a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133103268\" class=\"definition\">\n<dt id=\"fs-id1165133103274\"><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">coefficient matrix<\/span><\/dt>\n<\/dl>\n<dl class=\"definition\">\n<dt id=\"fs-id1165131866861\">\n<\/dt>\n<dd>a matrix that contains only the coefficients from a system of equations<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>column<\/dt>\n<dd>a set of numbers aligned vertically in a matrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1674058\" class=\"definition\">\n<dt>Cramer\u2019s Rule<\/dt>\n<dd id=\"fs-id1674063\">a method for solving systems of equations that have the same number of equations as variables using determinants<\/dd>\n<\/dl>\n<dl id=\"fs-id1674068\" class=\"definition\">\n<dt>determinant<\/dt>\n<dd id=\"fs-id1674074\">a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135639171\" class=\"definition\">\n<dt>element<\/dt>\n<dd id=\"fs-id1165135639177\">a number in a matrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134245049\" class=\"definition\">\n<dt>Gaussian elimination<\/dt>\n<dd id=\"fs-id1165134245055\">using elementary row operations to obtain a matrix in row-echelon form<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134179616\" class=\"definition\">\n<dt>identity matrix<\/dt>\n<dd id=\"fs-id1165134179622\">a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135639180\" class=\"definition\">\n<dt>matrix<\/dt>\n<dd id=\"fs-id1165137937501\">a rectangular array of numbers<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134179626\" class=\"definition\">\n<dt>multiplicative inverse of a matrix<\/dt>\n<dd id=\"fs-id1165135528440\">a matrix that, when multiplied by the original, equals the identity matrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134033250\" class=\"definition\">\n<dt>row<\/dt>\n<dd id=\"fs-id1165137937510\">a set of numbers aligned horizontally in a matrix<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134493421\" class=\"definition\">\n<dt>row-echelon form<\/dt>\n<dd id=\"fs-id1165134486699\">after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134486705\" class=\"definition\">\n<dt>row-equivalent<\/dt>\n<dd>two matrices [latex]A[\/latex] and [latex]B[\/latex] are row-equivalent if one can be obtained from the other by performing basic row operations<\/dd>\n<\/dl>\n<dl id=\"fs-id1165137456553\" class=\"definition\">\n<dt>row operations<\/dt>\n<dd id=\"fs-id1165132961357\">adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135199312\" class=\"definition\">\n<dt>scalar multiple<\/dt>\n<dd id=\"fs-id1165135199316\">an entry of a matrix that has been multiplied by a scalar<\/dd>\n<\/dl>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":514,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/696"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":10,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/696\/revisions"}],"predecessor-version":[{"id":5387,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/696\/revisions\/5387"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/514"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/696\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=696"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=696"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=696"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=696"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}