Question

perform the matrix multiplication, if possible.
\

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

\cdot \

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

options:
\

$$\begin{bmatrix} -4 & 12 & -4 \\ 1 & -9 & 1 \\end{bmatrix}$$

(yellow),
\

$$\begin{bmatrix} -4 & -12 & 4 \\ 1 & -3 & -1 \\end{bmatrix}$$

(purple),
\

$$\begin{bmatrix} -1 & -6 & 2 \\ -1 & -3 & 1 \\end{bmatrix}$$

(orange),
cannot be multiplied, dimensions not compatible. (teal)

Explanation:

Step1: Verify matrix compatibility

First matrix: $A =

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

$ (dimensions $2 \times 2$)
Second matrix: $B =

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

$ (dimensions $2 \times 3$)
Matrix multiplication is possible because the number of columns in $A$ (2) equals the number of rows in $B$ (2). The resulting matrix will be $2 \times 3$.

Step2: Calculate first row of product

First row, first column: $(0)(0) + (4)(-1) = -4$
First row, second column: $(0)(-3) + (4)(3) = 12$
First row, third column: $(0)(0) + (4)(-1) = -4$

Step3: Calculate second row of product

Second row, first column: $(2)(0) + (-1)(-1) = 1$
Second row, second column: $(2)(-3) + (-1)(3) = -9$
Second row, third column: $(2)(0) + (-1)(-1) = 1$

Answer:

$$\begin{bmatrix} -4 & 12 & -4 \\ 1 & -9 & 1 \end{bmatrix}$$