Question

section 6.3: problem 10
(10 points)
find the product. if it is not possible to perform this operation, explain.
\\(\

$$\begin{bmatrix}-1 & -8 & -4\\\\ -6 & -6 & 2\\\\ -8 & -6 & 7\\end{bmatrix}$$

\

$$\begin{bmatrix}-1 & -9 & -4\\\\ 1 & 9 & 5\\\\ 3 & -6 & -5\\end{bmatrix}$$

\\)
select the correct choice below, and, if necessary, fill in the answer boxes to complete your choice.
\\(\bigcirc\\) a. \\(\

$$\begin{bmatrix}-1 & -8 & -4\\\\ -6 & -6 & 2\\\\ -8 & -6 & 7\\end{bmatrix}$$

\

$$\begin{bmatrix}-1 & -9 & -4\\\\ 1 & 9 & 5\\\\ 3 & -6 & -5\\end{bmatrix}$$

= \

$$\begin{bmatrix}\\square & \\square & \\square\\\\ \\square & \\square & \\square\\\\ \\square & \\square & \\square\\end{bmatrix}$$

\\)
\\(\bigcirc\\) b. the operation is not defined because the number of columns in the first matrix is not the same as the number of rows in the second matrix.
\\(\bigcirc\\) c. the operation is not defined because the number of rows in the first matrix is not the same as the number of columns in the second matrix.
\\(\bigcirc\\) d. the product of two matrices of different sizes is not defined.
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Explanation:

Step1: Check matrix dimensions

First matrix: \( 3 \times 3 \) (3 rows, 3 columns). Second matrix: \( 3 \times 3 \) (3 rows, 3 columns). For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, both have 3 columns (first) and 3 rows (second), so multiplication is possible.

Step2: Calculate element (1,1)

First row of first matrix: \([-1, -8, -4]\), first column of second matrix: \([-1, 1, 3]\).
Product: \((-1)(-1) + (-8)(1) + (-4)(3) = 1 - 8 - 12 = -19\).

Step3: Calculate element (1,2)

First row of first matrix: \([-1, -8, -4]\), second column of second matrix: \([-9, 9, -6]\).
Product: \((-1)(-9) + (-8)(9) + (-4)(-6) = 9 - 72 + 24 = -39\).

Step4: Calculate element (1,3)

First row of first matrix: \([-1, -8, -4]\), third column of second matrix: \([-4, 5, -5]\).
Product: \((-1)(-4) + (-8)(5) + (-4)(-5) = 4 - 40 + 20 = -16\).

Step5: Calculate element (2,1)

Second row of first matrix: \([-6, -6, 2]\), first column of second matrix: \([-1, 1, 3]\).
Product: \((-6)(-1) + (-6)(1) + (2)(3) = 6 - 6 + 6 = 6\).

Step6: Calculate element (2,2)

Second row of first matrix: \([-6, -6, 2]\), second column of second matrix: \([-9, 9, -6]\).
Product: \((-6)(-9) + (-6)(9) + (2)(-6) = 54 - 54 - 12 = -12\).

Step7: Calculate element (2,3)

Second row of first matrix: \([-6, -6, 2]\), third column of second matrix: \([-4, 5, -5]\).
Product: \((-6)(-4) + (-6)(5) + (2)(-5) = 24 - 30 - 10 = -16\).

Step8: Calculate element (3,1)

Third row of first matrix: \([-8, -6, 7]\), first column of second matrix: \([-1, 1, 3]\).
Product: \((-8)(-1) + (-6)(1) + (7)(3) = 8 - 6 + 21 = 23\).

Step9: Calculate element (3,2)

Third row of first matrix: \([-8, -6, 7]\), second column of second matrix: \([-9, 9, -6]\).
Product: \((-8)(-9) + (-6)(9) + (7)(-6) = 72 - 54 - 42 = -24\).

Step10: Calculate element (3,3)

Third row of first matrix: \([-8, -6, 7]\), third column of second matrix: \([-4, 5, -5]\).
Product: \((-8)(-4) + (-6)(5) + (7)(-5) = 32 - 30 - 35 = -33\).

Answer:

A. \(

$$\begin{bmatrix}-1 & -8 & -4 \\ -6 & -6 & 2 \\ -8 & -6 & 7\end{bmatrix}$$

$$\begin{bmatrix}-1 & -9 & -4 \\ 1 & 9 & 5 \\ 3 & -6 & -5\end{bmatrix}$$

=

$$\begin{bmatrix}-19 & -39 & -16 \\ 6 & -12 & -16 \\ 23 & -24 & -33\end{bmatrix}$$

\)