Sure, let's calculate the matrix product [tex]\( AB \)[/tex] for the given matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Given:
[tex]\[
A = \begin{bmatrix} 1 & 7 \\ -3 & 2 \end{bmatrix}
\][/tex]
[tex]\[
B = \begin{bmatrix} 6 & 0 \\ 1 & -2 \end{bmatrix}
\][/tex]
To find the product [tex]\( AB \)[/tex], we need to perform matrix multiplication. The element in the first row and first column of [tex]\( AB \)[/tex] is calculated by taking the dot product of the first row of [tex]\( A \)[/tex] and the first column of [tex]\( B \)[/tex]. Similarly, we will calculate the other elements.
1. Element (1,1) of [tex]\( AB \)[/tex]:
[tex]\[
(1 \times 6) + (7 \times 1) = 6 + 7 = 13
\][/tex]
2. Element (1,2) of [tex]\( AB \)[/tex]:
[tex]\[
(1 \times 0) + (7 \times -2) = 0 - 14 = -14
\][/tex]
3. Element (2,1) of [tex]\( AB \)[/tex]:
[tex]\[
(-3 \times 6) + (2 \times 1) = -18 + 2 = -16
\][/tex]
4. Element (2,2) of [tex]\( AB \)[/tex]:
[tex]\[
(-3 \times 0) + (2 \times -2) = 0 - 4 = -4
\][/tex]
Now, putting all these elements together, we get:
[tex]\[
AB = \begin{bmatrix} 13 & -14 \\ -16 & -4 \end{bmatrix}
\][/tex]
So, the product [tex]\( AB \)[/tex] is:
[tex]\[
\begin{bmatrix} 13 & -14 \\ -16 & -4 \end{bmatrix}
\][/tex]
