To determine whether the matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are inverses of each other, we need to follow these steps:
1. Matrix Multiplication: First, compute the product of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. For two matrices to be inverses, their product must equal the identity matrix.
Given:
[tex]\[ A = \begin{pmatrix}
-2 & -3 \\
2 & -2
\end{pmatrix} \][/tex]
and
[tex]\[ B = \begin{pmatrix}
1 & -1 \\
-1 & 2
\end{pmatrix} \][/tex]
2. Compute [tex]\( AB \)[/tex]:
[tex]\[ AB = \begin{pmatrix}
-2 & -3 \\
2 & -2
\end{pmatrix} \cdot
\begin{pmatrix}
1 & -1 \\
-1 & 2
\end{pmatrix} \][/tex]
To compute [tex]\( AB \)[/tex], take the dot product of rows of [tex]\( A \)[/tex] with columns of [tex]\( B \)[/tex]:
- The element in the first row and first column of [tex]\( AB \)[/tex]:
[tex]\[ (-2 \cdot 1) + (-3 \cdot -1) = -2 + 3 = 1 \][/tex]
- The element in the first row and second column of [tex]\( AB \)[/tex]:
[tex]\[ (-2 \cdot -1) + (-3 \cdot 2) = 2 - 6 = -4 \][/tex]
- The element in the second row and first column of [tex]\( AB \)[/tex]:
[tex]\[ (2 \cdot 1) + (-2 \cdot -1) = 2 + 2 = 4 \][/tex]
- The element in the second row and second column of [tex]\( AB \)[/tex]:
[tex]\[ (2 \cdot -1) + (-2 \cdot 2) = -2 - 4 = -6 \][/tex]
Thus, the product [tex]\( AB \)[/tex] is:
[tex]\[ AB = \begin{pmatrix}
1 & -4 \\
4 & -6
\end{pmatrix} \][/tex]
3. Compare with Identity Matrix: The identity matrix for 2x2 matrices is:
[tex]\[ I = \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} \][/tex]
4. Conclusion: Compare [tex]\( AB \)[/tex] with the identity matrix [tex]\( I \)[/tex]. Since:
[tex]\[ \begin{pmatrix}
1 & -4 \\
4 & -6
\end{pmatrix} \neq \begin{pmatrix}
1 & 0 \\
0 & 1
\end{pmatrix} \][/tex]
The product [tex]\( AB \)[/tex] is not equal to the identity matrix. Therefore, matrices [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are not inverse matrices.
Answer: No
