Find the following matrix​ product, if it exists.
left bracket Start 3 By 3 Matrix 1st Row 1st Column 1 2nd Column 0 3rd Column 0 2nd Row 1st Column 0 2nd Column 1 3rd Column 0 3rd Row 1st Column 0 2nd Column 0 3rd Column 1 EndMatrix right bracket 1	0	0	0	1	0	0	0	1  left bracket Start 3 By 3 Matrix 1st Row 1st Column negative 3 2nd Column 5 3rd Column negative 9 2nd Row 1st Column negative 5 2nd Column negative 8 3rd Column negative 7 3rd Row 1st Column 0 2nd Column negative 1 3rd Column 2 EndMatrix right bracket −3	5	−9	−5	−8	−7	0	−1	2

To find the matrix product, we are given two \(3 \times 3\) matrices:

\[
A = \begin{bmatrix} 
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 
\end{bmatrix}
\quad \text{and} \quad
B = \begin{bmatrix} 
-3 & 5 & -9 \\
-5 & -8 & -7 \\
0 & -1 & 2 
\end{bmatrix}
\]

The matrix \(A\) is the identity matrix, which means when we multiply it with any other matrix, the result will be the second matrix itself. This is because the identity matrix leaves the other matrix unchanged during multiplication.

Thus, the product \(A \times B\) is:

\[
A \times B = \begin{bmatrix} 
-3 & 5 & -9 \\
-5 & -8 & -7 \\
0 & -1 & 2 
\end{bmatrix}
\]

So, the matrix product is:

\[
\begin{bmatrix} 
-3 & 5 & -9 \\
-5 & -8 & -7 \\
0 & -1 & 2 
\end{bmatrix}
\]

This is the result.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you multiply two general \(3 \times 3\) matrices?
2. What is the role of the identity matrix in matrix multiplication?
3. How does matrix multiplication differ from scalar multiplication?
4. Can matrix multiplication be commutative?
5. What are the properties of the identity matrix?

**Tip**: Always remember that multiplying any matrix by the identity matrix will leave the original matrix unchanged.

Find the following matrix product, if it exists. 
[ [1, 0, 0], [0, 1, 0], [0, 0, 1] ] × [ [-3, 5, -9], [-5, -8, -7], [0, -1, 2] ]

Explore how to find the product of a 3x3 identity matrix and another 3x3 matrix. This problem demonstrates the properties of matrix multiplication and the role of the identity matrix, suitable for high school students learning linear algebra concepts.

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