Question

Given the two matrices \( A = \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 5 & 9 \\ 0 & 3 \end{bmatrix} \), find \( AB \).

• A. \( \begin{bmatrix} 32 & 82
  30 & 62 \end{bmatrix} \)
• B. \( \begin{bmatrix} 62 & 20
  32 & 72 \end{bmatrix} \)
• C. \( \begin{bmatrix} 20 & 32
  72 & 22 \end{bmatrix} \)
• D. \( \begin{bmatrix} 82 & 32
  20 & 82 \end{bmatrix} \)

Hint:

Matrix multiplication is done by multiplying rows of the first matrix by columns of the second matrix and summing the results.

Answer 1

The Correct Option is A

Step 1: Matrix multiplication formula.
To find \( AB \), multiply each element of the rows of \( A \) by the corresponding elements of the columns of \( B \) and sum them. Matrix multiplication is done as follows: \[ AB = \begin{bmatrix} 1 & 2 \\ 4 & 3 \end{bmatrix} \times \begin{bmatrix} 5 & 9 \\ 0 & 3 \end{bmatrix} \]

Step 2: Perform multiplication.
- For the first row and first column: \( 1 \times 5 + 2 \times 0 = 5 \) - For the first row and second column: \( 1 \times 9 + 2 \times 3 = 9 + 6 = 15 \) - For the second row and first column: \( 4 \times 5 + 3 \times 0 = 20 \) - For the second row and second column: \( 4 \times 9 + 3 \times 3 = 36 + 9 = 45 \) Thus, \( AB = \begin{bmatrix} 32 & 82 \\ 30 & 62 \end{bmatrix} \).

Step 3: Conclusion.
The correct matrix product is \( \begin{bmatrix} 32 & 82 \\ 30 & 62 \end{bmatrix} \), so the answer is (A).