Problem 1. Matrix Operations. 10 points.
Let $A$ and $B$ be matrices defined as
\[
A=\left[\begin{array}{cc}
-1 & 2 \\
-5 & 10
\end{array}\right] \quad \text { and } \quad B=\left[\begin{array}{ll}
5 & 11 \\
4 & 12
\end{array}\right]
\]

Find a matrix $X$ which satisfies the following matrix equation
\[
5 X-4 A=-3 B .
\]

Step 1 ：Rearrange the given matrix equation to solve for X: \(X = \frac{{4A + 3B}}{5}\)

Step 2 ：Substitute the given matrices A and B into this equation: \(A = \begin{bmatrix} -1 & 2 \\ -5 & 10 \end{bmatrix}\) and \(B = \begin{bmatrix} 5 & 11 \\ 4 & 12 \end{bmatrix}\)

Step 3 ：Calculate 4A: \(4A = 4 * \begin{bmatrix} -1 & 2 \\ -5 & 10 \end{bmatrix} = \begin{bmatrix} -4 & 8 \\ -20 & 40 \end{bmatrix}\)

Step 4 ：Calculate 3B: \(3B = 3 * \begin{bmatrix} 5 & 11 \\ 4 & 12 \end{bmatrix} = \begin{bmatrix} 15 & 33 \\ 12 & 36 \end{bmatrix}\)

Step 5 ：Add the two matrices 4A and 3B: \(4A + 3B = \begin{bmatrix} -4 & 8 \\ -20 & 40 \end{bmatrix} + \begin{bmatrix} 15 & 33 \\ 12 & 36 \end{bmatrix} = \begin{bmatrix} 11 & 41 \\ -8 & 76 \end{bmatrix}\)

Step 6 ：Divide the result by 5 to find X: \(X = \frac{{4A + 3B}}{5} = \frac{{\begin{bmatrix} 11 & 41 \\ -8 & 76 \end{bmatrix}}}{5} = \begin{bmatrix} 2.2 & 8.2 \\ -1.6 & 15.2 \end{bmatrix}\)

Step 7 ：\(\boxed{X = \begin{bmatrix} 2.2 & 8.2 \\ -1.6 & 15.2 \end{bmatrix}}\)