Question

find the product of the following two matrices.
$\

$$\begin{bmatrix}-2&3\\\\1&1\\\\-4&0\\\\0&3\\\\0&4\\end{bmatrix}$$

\

$$\begin{bmatrix}-1&1\\\\3&-1\\end{bmatrix}$$

$

Explanation:

Step1: Recall matrix multiplication rule

For matrices $A_{m \times n}$ and $B_{n \times p}$, product $C_{m \times p}$ has $C_{ij}=\sum_{k=1}^n A_{ik}B_{kj}$.

Step2: Calculate first row of product

$$\begin{bmatrix} (-2)(-1)+(3)(3) & (-2)(1)+(3)(-1) \end{bmatrix} =\begin{bmatrix} 2+9 & -2-3 \end{bmatrix} =\begin{bmatrix} 11 & -5 \end{bmatrix}$$

Step3: Calculate second row of product

$$\begin{bmatrix} (1)(-1)+(1)(3) & (1)(1)+(1)(-1) \end{bmatrix} =\begin{bmatrix} -1+3 & 1-1 \end{bmatrix} =\begin{bmatrix} 2 & 0 \end{bmatrix}$$

Step4: Calculate third row of product

$$\begin{bmatrix} (-4)(-1)+(0)(3) & (-4)(1)+(0)(-1) \end{bmatrix} =\begin{bmatrix} 4+0 & -4+0 \end{bmatrix} =\begin{bmatrix} 4 & -4 \end{bmatrix}$$

Step5: Calculate fourth row of product

$$\begin{bmatrix} (0)(-1)+(3)(3) & (0)(1)+(3)(-1) \end{bmatrix} =\begin{bmatrix} 0+9 & 0-3 \end{bmatrix} =\begin{bmatrix} 9 & -3 \end{bmatrix}$$

Step6: Calculate fifth row of product

$$\begin{bmatrix} (0)(-1)+(4)(3) & (0)(1)+(4)(-1) \end{bmatrix} =\begin{bmatrix} 0+12 & 0-4 \end{bmatrix} =\begin{bmatrix} 12 & -4 \end{bmatrix}$$

Answer:

$$\begin{bmatrix} 11 & -5 \\ 2 & 0 \\ 4 & -4 \\ 9 & -3 \\ 12 & -4 \end{bmatrix}$$