https://file.aiquickdraw.com/mathbot/file/172895989168884h57wpk.jpg

We are tasked with finding the matrix product of the two given matrices, if it exists.

The matrices are:

\[
\mathbf{A} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\quad \text{and} \quad
\mathbf{B} =
\begin{pmatrix}
-8 & -1 & 9 \\
1 & -9 & -5 \\
0 & 2 & 6
\end{pmatrix}
\]

### Step 1: Matrix Multiplication Rule
To find the product \(\mathbf{A} \cdot \mathbf{B}\), we multiply each element of row \(i\) from matrix \(\mathbf{A}\) by each element of column \(j\) from matrix \(\mathbf{B}\) and sum the results. In this case, matrix \(\mathbf{A}\) is the identity matrix, so multiplying any matrix by the identity matrix results in the same matrix.

### Step 2: Calculate the product
Since \(\mathbf{A}\) is the identity matrix, the product \(\mathbf{A} \cdot \mathbf{B}\) will be the same as \(\mathbf{B}\):

\[
\mathbf{A} \cdot \mathbf{B} =
\begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\cdot
\begin{pmatrix}
-8 & -1 & 9 \\
1 & -9 & -5 \\
0 & 2 & 6
\end{pmatrix}
=
\begin{pmatrix}
-8 & -1 & 9 \\
1 & -9 & -5 \\
0 & 2 & 6
\end{pmatrix}
\]

### Final Answer:
The product exists and is:

\[
\begin{pmatrix}
-8 & -1 & 9 \\
1 & -9 & -5 \\
0 & 2 & 6
\end{pmatrix}
\]

So, the correct choice is **A**. 

Would you like further details on matrix multiplication? Here are some related questions for further exploration:

1. What is the effect of multiplying by the identity matrix on any matrix?
2. How would the result change if \(\mathbf{A}\) was not an identity matrix?
3. What conditions are necessary for two matrices to be multiplicable?
4. Can the order of matrix multiplication change the result? Why or why not?
5. How does matrix multiplication apply to transformations in geometry?

**Tip:** Always verify the dimensions of matrices before attempting to multiply them, to ensure the product exists.

Find the following matrix product, if it exists. Matrix A is a 3x3 identity matrix, and matrix B is a 3x3 matrix with elements: 
B = 
[[-8, -1, 9], 
[1, -9, -5], 
[0, 2, 6]].

This page provides a detailed explanation of matrix multiplication involving a 3x3 identity matrix and another 3x3 matrix. Understand the properties of the identity matrix and how to calculate matrix products. Ideal for high school students in Grades 10-12.

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