To determine whether Matrix A and Matrix B are inverses, we need to check if the product of A and B results in the identity matrix [tex]\( I \)[/tex].
Given:
[tex]\[
A = \begin{bmatrix} 2 & -6 \\ 1 & -1 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} -3 & 2 \\ 2 & -1 \end{bmatrix}
\][/tex]
We need to compute the product [tex]\( AB \)[/tex]:
[tex]\[
AB = \begin{bmatrix} 2 & -6 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} -3 & 2 \\ 2 & -1 \end{bmatrix}
\][/tex]
To calculate [tex]\( AB \)[/tex], we perform the following matrix multiplication:
[tex]\[
AB = \begin{bmatrix}
(2 -3 + -6 2) & (2 2 + -6 -1) \\
(1 -3 + -1 2) & (1 2 + -1 -1)
\end{bmatrix}
\][/tex]
Calculating each element:
- Top left: [tex]\( 2 -3 + -6 2 = -6 + (-12) = -18 \)[/tex]
- Top right: [tex]\( 2 2 + -6 -1 = 4 + 6 = 10 \)[/tex]
- Bottom left: [tex]\( 1 -3 + -1 2 = -3 + -2 = -5 \)[/tex]
- Bottom right: [tex]\( 1 2 + -1 -1 = 2 + 1 = 3 \)[/tex]
So, we get:
[tex]\[
AB = \begin{bmatrix} -18 & 10 \\ -5 & 3 \end{bmatrix}
\][/tex]
To verify whether A and B are inverses, the product matrix AB should be the identity matrix [tex]\( I \)[/tex]:
[tex]\[
I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\][/tex]
Comparing [tex]\( AB \)[/tex] with the identity matrix [tex]\( I \)[/tex]:
- [tex]\( AB \)[/tex] [tex]\( \neq \)[/tex] [tex]\( I \)[/tex]
Therefore, the result shows that [tex]\( AB \)[/tex] is not equal to the identity matrix. Hence, Matrix A and Matrix B are not inverse matrices.
The answer is: False
