https://file.aiquickdraw.com/mathbot/file/1726335672744ovbqhr9x.jpg

The given equation is:

\[
AX + B = C
\]
where:

\[
A = \begin{pmatrix} 3 & 0 & 5 \\ 0 & 1 & 3 \\ -1 & 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 3 \\ 0 & 1 \\ 2 & -1 \end{pmatrix}, \quad C = \begin{pmatrix} 0 & 3 \\ 2 & -1 \\ 4 & 5 \end{pmatrix}.
\]

We are tasked with solving for \( X \), where \( X \) is a matrix.

### Steps to solve:

1. Start with the equation:
   \[
   AX + B = C
   \]

2. Subtract \( B \) from both sides:
   \[
   AX = C - B
   \]

3. Compute \( C - B \), then solve for \( X \) by multiplying both sides by \( A^{-1} \) (the inverse of matrix \( A \)).

Let me calculate the intermediate steps for you.The solution for the matrix \( X \) is:

\[
X = \begin{pmatrix} -1.5 & 5 \\ -0.1 & 7 \\ 0.7 & -3 \end{pmatrix}
\]

This matrix satisfies the equation \( AX + B = C \).

Would you like any further details or have questions? Here are some related questions for further exploration:

1. How do we calculate the inverse of a matrix?
2. What are the properties of matrix multiplication?
3. Can you solve the system using Gaussian elimination instead of matrix inversion?
4. What conditions make a matrix invertible?
5. How do errors in matrix elements affect the final solution in real-world applications?

**Tip:** When performing matrix operations, always ensure that the matrix is invertible before calculating the inverse.

Solve for the matrix X in the equation AX + B = C, where A, B, and C are given matrices.

Solve the matrix equation AX + B = C by using matrix operations. The problem involves matrix subtraction and matrix inversion. Learn how to calculate the inverse of a matrix and solve for X in a step-by-step process, suitable for undergraduate and advanced high school students.

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