Matrices and Matrix Operations: Cheat Sheet

Essential Concepts

Matrices and Matrix Operations

Matrix Basics

  • A matrix is a rectangular array of numbers arranged in rows and columns
  • Dimensions are written as [latex]m \times n[/latex], where [latex]m[/latex] = number of rows and [latex]n[/latex] = number of columns
  • 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

Matrix Addition and Subtraction

  • Matrices must have equal dimensions to be added or subtracted
  • Add or subtract corresponding entries from each matrix

Scalar Multiplication

  • Multiply every entry in the matrix by the scalar (constant)

Matrix Multiplication

  • Only possible when inner dimensions match: the number of columns in the first matrix must equal the number of rows in the second matrix
  • 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]
  • To find entry [latex]c_{ij}[/latex] of product matrix [latex]C[/latex]:
    • Multiply each entry in row [latex]i[/latex] of [latex]A[/latex] by corresponding entry in column [latex]j[/latex] of [latex]B[/latex]
    • Sum all products: [latex]c_{ij} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{in}b_{nj}[/latex]
  • Matrix multiplication is not commutative: [latex]AB \neq BA[/latex] in general

Solving Systems with Gaussian Elimination

Augmented Matrices

  • An augmented matrix represents a system of equations using coefficients and constants
  • Written as [latex][A|B][/latex] where [latex]A[/latex] contains coefficients and [latex]B[/latex] contains constants
  • 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]

Row Operations Three types of row operations can be performed:

  1. Multiply a row by a nonzero constant
  2. Add a multiple of one row to another row
  3. Interchange (swap) two rows

Row-Echelon Form A matrix is in row-echelon form when:

  • All rows of zeros (if any) are at the bottom
  • The first nonzero entry in each row (called a leading entry) is to the right of the leading entry in the row above
  • All entries below a leading entry are zeros

Gaussian Elimination Process

  1. Write the system as an augmented matrix
  2. Use row operations to obtain row-echelon form
  3. Back-substitute to find solutions, starting from the bottom row

Solving Systems with Inverses

Identity Matrix

  • Denoted [latex]I[/latex], has 1’s on the main diagonal and 0’s elsewhere
  • Property: [latex]AI = IA = A[/latex] for any compatible matrix [latex]A[/latex]
  • [latex]2 \times 2[/latex] identity: [latex]\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}\right][/latex]

Inverse Matrix

  • The inverse of matrix [latex]A[/latex] is denoted [latex]A^{-1}[/latex]
  • Property: [latex]AA^{-1} = A^{-1}A = I[/latex]
  • Not all matrices have inverses; a matrix with an inverse is called invertible
  • If [latex]\det(A) = 0[/latex], the matrix has no inverse

Finding the Inverse of a [latex]2 \times 2[/latex] Matrix

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]

Finding the Inverse Using Row Operations

  1. Form the augmented matrix [latex][A|I][/latex]
  2. Use row operations to transform the left side into [latex]I[/latex]
  3. The right side becomes [latex]A^{-1}[/latex]: [latex][A|I] \rightarrow [I|A^{-1}][/latex]
  4. If [latex]A[/latex] cannot be transformed to [latex]I[/latex], then [latex]A[/latex] has no inverse

Solving Systems Using Inverses

  1. Write the system as [latex]AX = B[/latex], where [latex]A[/latex] = coefficient matrix, [latex]X[/latex] = variable matrix, [latex]B[/latex] = constant matrix
  2. Multiply both sides by [latex]A^{-1}[/latex]: [latex]A^{-1}AX = A^{-1}B[/latex]
  3. Simplify: [latex]IX = A^{-1}B[/latex], so [latex]X = A^{-1}B[/latex]

Solving Systems with Cramer’s Rule

Determinants

  • 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]
  • For a [latex]3 \times 3[/latex] matrix, augment with the first two columns and use the diagonal method:
    • Add the three products of diagonals going from upper-left to lower-right
    • Subtract the three products of diagonals going from lower-left to upper-right

Cramer’s Rule

For the system [latex]AX = B[/latex]:

  • [latex]D = \det(A)[/latex] (determinant of coefficient matrix)
  • [latex]D_x = \det[/latex] of matrix with [latex]x[/latex]-column replaced by constant column [latex]B[/latex]
  • [latex]D_y = \det[/latex] of matrix with [latex]y[/latex]-column replaced by constant column [latex]B[/latex]
  • Solutions: [latex]x = \frac{D_x}{D}[/latex], [latex]y = \frac{D_y}{D}[/latex] (provided [latex]D \neq 0[/latex])

Key Equations

Identity matrix for a [latex]2\text{}\times \text{}2[/latex] matrix [latex]{I}_{2}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right][/latex]
Identity matrix for a [latex]\text{3}\text{}\times \text{}3[/latex] matrix [latex]{I}_{3}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right][/latex]
Multiplicative inverse of a [latex]2\text{}\times \text{}2[/latex] matrix [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]

Glossary

augmented matrix
a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets
coefficient matrix
a matrix that contains only the coefficients from a system of equations
column
a set of numbers aligned vertically in a matrix
Cramer’s Rule
a method for solving systems of equations that have the same number of equations as variables using determinants
determinant
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
element
a number in a matrix
Gaussian elimination
using elementary row operations to obtain a matrix in row-echelon form
identity matrix
a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra
matrix
a rectangular array of numbers
multiplicative inverse of a matrix
a matrix that, when multiplied by the original, equals the identity matrix
row
a set of numbers aligned horizontally in a matrix
row-echelon form
after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal
row-equivalent
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
row operations
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
scalar multiple
an entry of a matrix that has been multiplied by a scalar