Question

select the correct answer.
consider matrices a and b.
$a = \

$$\begin{bmatrix}-1 & 4 \\\\ 0 & 5\\end{bmatrix}$$

$, $b = \

$$\begin{bmatrix}7 & 9 & 2 \\\\ 2 & 0 & -3\\end{bmatrix}$$

$
what is the product of these matrices, $ab$?
a. $\

$$\begin{bmatrix}1 & -23 \\\\ 10 & -15\\end{bmatrix}$$

$
b. these two matrices cannot be multiplied.
c. $\

$$\begin{bmatrix}1 & 9 & -14 \\\\ -10 & 0 & 15\\end{bmatrix}$$

$
d. $\

$$\begin{bmatrix}1 & -9 & -14 \\\\ 10 & 0 & -15\\end{bmatrix}$$

$

Explanation:

Step1: Verify matrix multiplication validity

Matrix $A$ is $2 \times 2$, matrix $B$ is $2 \times 3$. Since the number of columns in $A$ equals the number of rows in $B$, multiplication is valid, resulting in a $2 \times 3$ matrix.

Step2: Calculate row1, column1 of $AB$

Multiply row1 of $A$ by column1 of $B$:
$(-1)(7) + (4)(2) = -7 + 8 = 1$

Step3: Calculate row1, column2 of $AB$

Multiply row1 of $A$ by column2 of $B$:
$(-1)(9) + (4)(0) = -9 + 0 = -9$

Step4: Calculate row1, column3 of $AB$

Multiply row1 of $A$ by column3 of $B$:
$(-1)(2) + (4)(-3) = -2 -12 = -14$

Step5: Calculate row2, column1 of $AB$

Multiply row2 of $A$ by column1 of $B$:
$(0)(7) + (5)(2) = 0 + 10 = 10$

Step6: Calculate row2, column2 of $AB$

Multiply row2 of $A$ by column2 of $B$:
$(0)(9) + (5)(0) = 0 + 0 = 0$

Step7: Calculate row2, column3 of $AB$

Multiply row2 of $A$ by column3 of $B$:
$(0)(2) + (5)(-3) = 0 -15 = -15$

Answer:

D.

$$\begin{bmatrix} 1 & -9 & -14 \\ 10 & 0 & -15 \end{bmatrix}$$