Question

If \( A = \begin{pmatrix} 1 & -4 \\ 3 & 3 \end{pmatrix} \) and \( B = \begin{pmatrix} -1 & 2 \\ 1 & 2 \end{pmatrix} \), then find \( AB \).

Hint:

For matrix multiplication, take the dot product of rows of the first matrix with columns of the second matrix.

Answer 1

Step 1: Matrix multiplication.

To find \( AB \), we multiply the two matrices \( A \) and \( B \). Matrix multiplication is done by taking the dot product of rows of \( A \) with the columns of \( B \). \[ AB = \begin{pmatrix} 1 & -4 \\ 3 & 3 \end{pmatrix} \times \begin{pmatrix} -1 & 2 \\ 1 & 2 \end{pmatrix} \] The element in the first row and first column of \( AB \) is: \[ 1 \times (-1) + (-4) \times 1 = -1 - 4 = -5 \] The element in the first row and second column of \( AB \) is: \[ 1 \times 2 + (-4) \times 2 = 2 - 8 = -6 \] The element in the second row and first column of \( AB \) is: \[ 3 \times (-1) + 3 \times 1 = -3 + 3 = 0 \] The element in the second row and second column of \( AB \) is: \[ 3 \times 2 + 3 \times 2 = 6 + 6 = 12 \] \[ AB = \begin{pmatrix} -5 & -6 \\ 0 & 12 \end{pmatrix} \]

Step 2: Conclusion. 

Thus, the product \( AB \) is: \[ AB = \begin{pmatrix} -5 & -6 \\ 0 & 12 \end{pmatrix} \]