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Examine the product of the two matrices to determine if each is the inverse of the other:

[tex]\[
\left[\begin{array}{rrr}
1 & -1 & 1 \\
0 & 3 & -1 \\
3 & 7 & 0
\end{array}\right]
\left[\begin{array}{rrr}
7 & 7 & -2 \\
-3 & -3 & 1 \\
-9 & -10 & 3
\end{array}\right]
\][/tex]

Are the matrices inverses of each other?

A. Yes
B. No



Answer :

GinnyAnswer
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

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