https://file.aiquickdraw.com/mathbot/file/1726828905851wfkh3qzw.jpg

The problem in the image is a matrix equation of the form:

\[
\begin{bmatrix} 
1 & -1 & 1 \\ 
2 & 3 & 0 \\ 
0 & 2 & -1 
\end{bmatrix}
X = 
\begin{bmatrix} 
4 & -1 & 7 & 9 & 10 \\ 
6 & 0 & -5 & 0 & 1 \\ 
5 & 7 & -9 & 4 & 1 
\end{bmatrix}
\]

You are asked to solve for the matrix \( X \).

### Steps to Solve:
This is a system of linear equations in matrix form \( AX = B \), where \( A \) is the 3x3 matrix on the left, and \( B \) is the 3x5 matrix on the right. To solve for \( X \), follow these steps:

1. **Multiply both sides by the inverse of \( A \)**:
   \[
   A^{-1} A X = A^{-1} B
   \]
   Since \( A^{-1} A = I \) (the identity matrix), this simplifies to:
   \[
   X = A^{-1} B
   \]

2. **Find the inverse of matrix \( A \)**.

3. **Multiply the inverse of \( A \) by matrix \( B \)** to get \( X \).

Let me calculate that for you.The matrix \( X \) is:

\[
X = \begin{bmatrix} 
21 & 18 & -1 & 39 & 32 \\ 
-12 & -12 & -1 & -26 & -21 \\ 
-29 & -31 & 7 & -56 & -43 
\end{bmatrix}
\]

This is the solution for the matrix equation.

Let me know if you'd like more details, or if you have any questions!

Here are 5 related questions you can explore:
1. How do you calculate the inverse of a matrix?
2. Can every matrix have an inverse? Why or why not?
3. What happens if a matrix is singular (non-invertible)?
4. How can matrix multiplication be used to solve systems of linear equations?
5. What are some real-life applications of matrix equations?

**Tip**: Always check if a matrix is invertible by verifying its determinant is non-zero.

Solve the following matrix equation for X: [ [1, -1, 1], [2, 3, 0], [0, 2, -1] ] * X = [ [4, -1, 7, 9, 10], [6, 0, -5, 0, 1], [5, 7, -9, 4, 1] ]

Learn how to solve the matrix equation [1, -1, 1; 2, 3, 0; 0, 2, -1] * X = [4, -1, 7, 9, 10; 6, 0, -5, 0, 1; 5, 7, -9, 4, 1] for matrix X using the inverse of the matrix A. This problem involves linear algebra concepts like matrix multiplication, inverse, and systems of equations. Suitable for students in Grades 10-12.

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