Matrix Multiplication Question

Solve the Matrix Equation

Given two matrices \( A \) and \( B \):

\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \]

Find the product \( A \times B \).

Solution

To find the product \( A \times B \), we perform matrix multiplication. For 2x2 matrices, the resulting matrix will also be 2x2. We calculate each element of the resulting matrix as follows:

  1. Element at (1,1):

    \[ (A \times B)\_{11} = (1 \times 5) + (2 \times 7) = 5 + 14 = 19 \]

  2. Element at (1,2):

    \[ (A \times B)\_{12} = (1 \times 6) + (2 \times 8) = 6 + 16 = 22 \]

  3. Element at (2,1):

    \[ (A \times B)\_{21} = (3 \times 5) + (4 \times 7) = 15 + 28 = 43 \]

  4. Element at (2,2):

    \[ (A \times B)\_{22} = (3 \times 6) + (4 \times 8) = 18 + 32 = 50 \]

Therefore, the product \( A \times B \) is:

\[ A \times B = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} \]

Verification

To verify our result, we can check that each element of the product matrix is correct:

  • (1,1): \( 1(5) + 2(7) = 5 + 14 = 19 \)
  • (1,2): \( 1(6) + 2(8) = 6 + 16 = 22 \)
  • (2,1): \( 3(5) + 4(7) = 15 + 28 = 43 \)
  • (2,2): \( 3(6) + 4(8) = 18 + 32 = 50 \)

This confirms that our calculation of \( A \times B \) is correct.