Question

perform the matrix multiplication, if possible.
\

$$\begin{bmatrix}5 & -3 \\2 & -5\\end{bmatrix}$$

\cdot \

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

(options:
\

$$\begin{bmatrix}29 & 24 & -2 \\23 & 21 & -16\\end{bmatrix}$$

,
\

$$\begin{bmatrix}29 & 24 & -2 \\7 & 9 & -24\\end{bmatrix}$$

,
\

$$\begin{bmatrix}29 & 24 & -22 \\23 & 21 & -24\\end{bmatrix}$$

,
cannot be multiplied, dimensions not compatible.)

Explanation:

Step1: Verify multiplication compatibility

The first matrix is $2 \times 2$, the second is $2 \times 3$. Since the number of columns in the first matrix equals the number of rows in the second, multiplication is possible, resulting in a $2 \times 3$ matrix.

Step2: Calculate first row elements

Compute each element of the resulting matrix's first row by taking the dot product of the first row of the first matrix with each column of the second matrix:
First element: $5 \times 4 + (-3) \times (-3) = 20 + 9 = 29$
Second element: $5 \times 3 + (-3) \times (-3) = 15 + 9 = 24$
Third element: $5 \times 2 + (-3) \times 4 = 10 - 12 = -2$

Step3: Calculate second row elements

Compute each element of the resulting matrix's second row by taking the dot product of the second row of the first matrix with each column of the second matrix:
First element: $2 \times 4 + (-5) \times (-3) = 8 + 15 = 23$
Second element: $2 \times 3 + (-5) \times (-3) = 6 + 15 = 21$
Third element: $2 \times 2 + (-5) \times 4 = 4 - 20 = -16$

Answer:

$$\begin{bmatrix} 29 & 24 & -2 \\ 23 & 21 & -16 \end{bmatrix}$$