To determine whether two matrices are inverses of each other, we need to check if their product is the identity matrix. We start with the given matrices:
[tex]\[
A = \left[\begin{array}{rrr}
1 & -1 & 1 \\
0 & 3 & -1 \\
3 & 7 & 0
\end{array}\right]
\][/tex]
[tex]\[
B = \left[\begin{array}{rrr}
7 & 7 & -2 \\
-3 & -3 & 1 \\
-9 & -10 & 3
\end{array}\right]
\][/tex]
The identity matrix for a 3x3 matrix is:
[tex]\[
I = \left[\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
\][/tex]
We need to compute the product [tex]\(AB\)[/tex]:
[tex]\[
AB = \left[\begin{array}{rrr}
1 & -1 & 1 \\
0 & 3 & -1 \\
3 & 7 & 0
\end{array}\right]
\left[\begin{array}{rrr}
7 & 7 & -2 \\
-3 & -3 & 1 \\
-9 & -10 & 3
\end{array}\right]
\][/tex]
After calculating the matrix product, we get:
[tex]\[
AB = \left[\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right]
\][/tex]
The product [tex]\(AB\)[/tex] is clearly the identity matrix [tex]\(I\)[/tex]. Therefore, the given matrices [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are inverses of each other.
So, the answer to the question "Are the matrices inverses of each other?" is:
Yes
