Determine whether B is the inverse of A.
A = 

2	1	−1	

5	3	−5
−1	−5	2
,   B = 

5	−1	5	

−3	4	5
−3	4	5
AB =

 	



Is B the inverse of A?

Yes
No    

To determine if matrix \( B \) is the inverse of matrix \( A \), we must check if the product \( AB = I \), where \( I \) is the identity matrix.

Let's begin by calculating the matrix multiplication of \( A \) and \( B \).

Given:

\[
A = \begin{pmatrix}
2 & 1 & -1 \\
5 & 3 & -5 \\
-1 & -5 & 2
\end{pmatrix}
\quad \text{and} \quad
B = \begin{pmatrix}
5 & -1 & 5 \\
-3 & 4 & 5 \\
-3 & 4 & 5
\end{pmatrix}
\]

The multiplication \( AB \) is carried out as follows:

\[
AB = \begin{pmatrix}
(2 \times 5 + 1 \times -3 + -1 \times -3) & (2 \times -1 + 1 \times 4 + -1 \times 4) & (2 \times 5 + 1 \times 5 + -1 \times 5) \\
(5 \times 5 + 3 \times -3 + -5 \times -3) & (5 \times -1 + 3 \times 4 + -5 \times 4) & (5 \times 5 + 3 \times 5 + -5 \times 5) \\
(-1 \times 5 + -5 \times -3 + 2 \times -3) & (-1 \times -1 + -5 \times 4 + 2 \times 4) & (-1 \times 5 + -5 \times 5 + 2 \times 5)
\end{pmatrix}
\]

Simplifying each entry:

First row:
\[
(10 + -3 + 3) = 10, \quad (-2 + 4 + -4) = -2, \quad (10 + 5 + -5) = 10
\]

Second row:
\[
(25 + -9 + 15) = 31, \quad (-5 + 12 + -20) = -13, \quad (25 + 15 + -25) = 15
\]

Third row:
\[
(-5 + 15 + -6) = 4, \quad (1 + -20 + 8) = -11, \quad (-5 + -25 + 10) = -20
\]

Thus, the product \( AB \) is:

\[
AB = \begin{pmatrix}
10 & -2 & 10 \\
31 & -13 & 15 \\
4 & -11 & -20
\end{pmatrix}
\]

Since \( AB \) is not the identity matrix (which would be \( I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)), matrix \( B \) is **not** the inverse of matrix \( A \).

### Conclusion:
**No**, \( B \) is not the inverse of \( A \).

Do you want any further details or explanations?

Here are 5 related questions for deeper exploration:
1. What is the general process for finding the inverse of a 3x3 matrix?
2. Can you compute the inverse of matrix \( A \) manually?
3. What are the properties of an identity matrix?
4. How would you determine if a matrix is singular?
5. How do you check if a matrix multiplication yields the identity matrix?

**Tip:** For two matrices \( A \) and \( B \) to be inverses, their product must equal the identity matrix both as \( AB \) and \( BA \).

Determine whether B is the inverse of A. A = [2, 1, -1; 5, 3, -5; -1, -5, 2], B = [5, -1, 5; -3, 4, 5; -3, 4, 5]. Is B the inverse of A?

Explore whether matrix B is the inverse of matrix A by calculating the matrix product AB and checking if it equals the identity matrix. This problem demonstrates key linear algebra concepts such as matrix multiplication and the conditions for matrix inverses. Suitable for Grades 11-12 and college-level students.

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