{"id":1372,"date":"2025-07-24T19:13:50","date_gmt":"2025-07-24T19:13:50","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1372"},"modified":"2026-03-23T18:13:35","modified_gmt":"2026-03-23T18:13:35","slug":"matrices-and-matrix-operations-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/matrices-and-matrix-operations-learn-it-2\/","title":{"raw":"Matrices and Matrix Operations: Learn It 2","rendered":"Matrices and Matrix Operations: Learn It 2"},"content":{"raw":"

Adding and Subtracting Matrices<\/h2>\r\nNow that we understand the basics of what a matrix is, let's move on to two important operations: adding and subtracting matrices.\r\n\r\nMatrices are made up of numbers arranged in rows and columns. Since these are numbers, we can add and subtract them just like we do with regular numbers. To add or subtract matrices, they must be of the same dimensions<\/strong>, meaning they must have the same number of rows and columns.\r\n\r\nWhen adding or subtracting matrices, we perform the operation on each corresponding element. For example, the number in row [latex]1[\/latex], column [latex]2[\/latex] of the first matrix must be added to or subtracted from the number in row [latex]1[\/latex], column [latex]2[\/latex] of the second matrix.\r\n\r\n
\r\n

adding and subtracting matrices<\/h3>\r\nGiven matrices [latex]A[\/latex] and [latex]B[\/latex] of like dimensions, addition and subtraction of [latex]A[\/latex] and [latex]B[\/latex] will produce matrix [latex]C[\/latex] or\u00a0matrix [latex]D[\/latex] of the same dimension.\r\n

[latex]A+B=C\\text{ such that }{a}_{ij}+{b}_{ij}={c}_{ij}[\/latex]<\/p>\r\n

[latex]A-B=D\\text{ such that }{a}_{ij}-{b}_{ij}={d}_{ij}[\/latex]<\/p>\r\nMatrix addition is commutative.\r\n

[latex]A+B=B+A[\/latex]<\/p>\r\nIt is also associative.\r\n

[latex]\\left(A+B\\right)+C=A+\\left(B+C\\right)[\/latex]<\/p>\r\n\r\n<\/section>

Given [latex]A[\/latex] and [latex]B:[\/latex]\r\n

[latex]A=\\left[\\begin{array}{rrr}\\hfill 2& \\hfill -10& \\hfill -2\\\\ \\hfill 14& \\hfill 12& \\hfill 10\\\\ \\hfill 4& \\hfill -2& \\hfill 2\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rrr}\\hfill 6& \\hfill 10& \\hfill -2\\\\ \\hfill 0& \\hfill -12& \\hfill -4\\\\ \\hfill -5& \\hfill 2& \\hfill -2\\end{array}\\right][\/latex]<\/p>\r\n\r\n

    \r\n \t
  1. Find the sum.<\/li>\r\n \t
  2. Find the difference.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"286263\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"286263\"]\r\n
      \r\n \t
    1. Add the corresponding entries.\r\n
      [latex]\\begin{array}{ll}\\hfill \\\\ A+B& =\\left[\\begin{array}{rrr}\\hfill 2& \\hfill -10& \\hfill -2\\\\ \\hfill 14& \\hfill 12& \\hfill 10\\\\ \\hfill 4& \\hfill -2& \\hfill 2\\end{array}\\right]+\\left[\\begin{array}{rrr}\\hfill 6& \\hfill 10& \\hfill -2\\\\ \\hfill 0& \\hfill -12& \\hfill -4\\\\ \\hfill -5& \\hfill 2& \\hfill -2\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 2+6& \\hfill -10+10& \\hfill -2 - 2\\\\ \\hfill 14+0& \\hfill 12 - 12& \\hfill 10 - 4\\\\ \\hfill 4 - 5& \\hfill -2+2& \\hfill 2 - 2\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 8& \\hfill 0& \\hfill -4\\\\ \\hfill 14& \\hfill 0& \\hfill 6\\\\ \\hfill -1& \\hfill 0& \\hfill 0\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n \t
    2. Subtract the corresponding entries.\r\n
      [latex]\\begin{array}{ll}\\hfill \\\\ A-B & =\\left[\\begin{array}{rrr}\\hfill 2& \\hfill -10& \\hfill -2\\\\ \\hfill 14& \\hfill 12& \\hfill 10\\\\ \\hfill 4& \\hfill -2& \\hfill 2\\end{array}\\right]-\\left[\\begin{array}{rrr}\\hfill 6& \\hfill 10& \\hfill -2\\\\ \\hfill 0& \\hfill -12& \\hfill -4\\\\ \\hfill -5& \\hfill 2& \\hfill -2\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill 2 - 6& \\hfill -10 - 10& \\hfill -2+2\\\\ \\hfill 14 - 0& \\hfill 12+12& \\hfill 10+4\\\\ \\hfill 4+5& \\hfill -2 - 2& \\hfill 2+2\\end{array}\\right]\\hfill \\\\ & =\\left[\\begin{array}{rrr}\\hfill -4& \\hfill -20& \\hfill 0\\\\ \\hfill 14& \\hfill 24& \\hfill 14\\\\ \\hfill 9& \\hfill -4& \\hfill 4\\end{array}\\right]\\hfill \\end{array}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>
      [ohm_question hide_question_numbers=1]321697[\/ohm_question]<\/section>
      [ohm_question hide_question_numbers=1]321698[\/ohm_question]<\/section>","rendered":"

      Adding and Subtracting Matrices<\/h2>\n

      Now that we understand the basics of what a matrix is, let’s move on to two important operations: adding and subtracting matrices.<\/p>\n

      Matrices are made up of numbers arranged in rows and columns. Since these are numbers, we can add and subtract them just like we do with regular numbers. To add or subtract matrices, they must be of the same dimensions<\/strong>, meaning they must have the same number of rows and columns.<\/p>\n

      When adding or subtracting matrices, we perform the operation on each corresponding element. For example, the number in row [latex]1[\/latex], column [latex]2[\/latex] of the first matrix must be added to or subtracted from the number in row [latex]1[\/latex], column [latex]2[\/latex] of the second matrix.<\/p>\n

      \n

      adding and subtracting matrices<\/h3>\n

      Given matrices [latex]A[\/latex] and [latex]B[\/latex] of like dimensions, addition and subtraction of [latex]A[\/latex] and [latex]B[\/latex] will produce matrix [latex]C[\/latex] or\u00a0matrix [latex]D[\/latex] of the same dimension.<\/p>\n

      [latex]A+B=C\\text{ such that }{a}_{ij}+{b}_{ij}={c}_{ij}[\/latex]<\/p>\n

      [latex]A-B=D\\text{ such that }{a}_{ij}-{b}_{ij}={d}_{ij}[\/latex]<\/p>\n

      Matrix addition is commutative.<\/p>\n

      [latex]A+B=B+A[\/latex]<\/p>\n

      It is also associative.<\/p>\n

      [latex]\\left(A+B\\right)+C=A+\\left(B+C\\right)[\/latex]<\/p>\n<\/section>\n

      Given [latex]A[\/latex] and [latex]B:[\/latex]<\/p>\n

      [latex]A=\\left[\\begin{array}{rrr}\\hfill 2& \\hfill -10& \\hfill -2\\\\ \\hfill 14& \\hfill 12& \\hfill 10\\\\ \\hfill 4& \\hfill -2& \\hfill 2\\end{array}\\right]\\text{ and }B=\\left[\\begin{array}{rrr}\\hfill 6& \\hfill 10& \\hfill -2\\\\ \\hfill 0& \\hfill -12& \\hfill -4\\\\ \\hfill -5& \\hfill 2& \\hfill -2\\end{array}\\right][\/latex]<\/p>\n

        \n
      1. Find the sum.<\/li>\n
      2. Find the difference.<\/li>\n<\/ol>\n