https://file.aiquickdraw.com/mathbot/file/1725975930555nqg4qjij.jpg

Let's analyze and solve the problems from the image you uploaded. There are two questions involving matrix operations.

### Problem 1:
You are given the matrices \( A \) and \( B \) as follows:

\[
A = \begin{pmatrix} 2 & -1 \\ -3 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 4 \\ -7 & -5 \end{pmatrix}
\]

The equation is \( AX = B \), and you are asked to find matrix \( X \).

To solve for \( X \), we use the equation:
\[
X = A^{-1} B
\]
We need to:
1. Compute the inverse of matrix \( A \).
2. Multiply \( A^{-1} \) with matrix \( B \).

### Problem 2:
You are asked to find matrix \( X \) that satisfies the following matrix equation:

\[
X \begin{pmatrix} 4 & -7 \\ 3 & -5 \end{pmatrix} = \begin{pmatrix} -3 & 10 \\ 7 & 8 \end{pmatrix}
\]

To find matrix \( X \), we multiply both sides by the inverse of the matrix on the right:
\[
X = \begin{pmatrix} -3 & 10 \\ 7 & 8 \end{pmatrix} \left( \begin{pmatrix} 4 & -7 \\ 3 & -5 \end{pmatrix} \right)^{-1}
\]

Let me calculate both steps for you.Here are the solutions to the matrix problems:

### Problem 1:
Given \( A \) and \( B \), the matrix \( X \) that satisfies \( AX = B \) is:

\[
X = \begin{pmatrix} 2 & 1 \\ -1 & -2 \end{pmatrix}
\]

### Problem 2:
For the equation \( X \begin{pmatrix} 4 & -7 \\ 3 & -5 \end{pmatrix} = \begin{pmatrix} -3 & 10 \\ 7 & 8 \end{pmatrix} \), the matrix \( X \) is:

\[
X = \begin{pmatrix} -15 & 19 \\ -59 & 81 \end{pmatrix}
\]

Let me know if you would like further details or explanations!

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Here are 5 related questions to help you deepen your understanding:

1. What is the determinant of matrix \( A \) in Problem 1?
2. How do you determine if a matrix has an inverse?
3. What are the properties of matrix multiplication?
4. Can you find the determinant of the matrix \( C \) in Problem 2?
5. What happens if a matrix has a determinant of zero?

**Tip**: When solving matrix equations, always check that the matrix you're inverting is non-singular by verifying that its determinant is non-zero!

1. Diketahui matriks A = (2 -1, -3 1) dan B = (5 4, -7 -5). Jika AX = B dengan X matriks berordo 2 x 2, tentukan matriks X! 2. Tentukan matriks X yang memenuhi X(4 -7, 3 -5) = (-3 10, 7 8).

Learn how to solve matrix equations involving matrix inversion and multiplication, including the equation AX = B and X(A) = B. The problems cover 2x2 matrices and detailed steps to find the inverse of a matrix.

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