https://file.aiquickdraw.com/mathbot/file/1727014284286jzne7qvs.jpg

In the image, we are given two matrices \( A \) and \( B \), and two tasks related to matrix equations:

Given matrices:
\[
A = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 4 \\ -7 & -8 \end{pmatrix}
\]

The problem asks to solve the following two cases:

1. Solve for \( x \) in the equation \( A \cdot x = B \)
2. Solve for \( X \) in the equation \( X \cdot A = B \)

### Part a: Solving \( A \cdot x = B \)

This represents a matrix equation of the form \( A \cdot x = B \). We want to find the matrix \( x \). Since \( A \) is a \( 2 \times 2 \) matrix, we expect \( x \) to also be a \( 2 \times 2 \) matrix. The solution is obtained by multiplying the inverse of \( A \) with \( B \), i.e.,

\[
x = A^{-1} \cdot B
\]

We will calculate this below.

### Part b: Solving \( X \cdot A = B \)

This equation is also a matrix equation, and we want to solve for matrix \( X \). To do so, we multiply both sides by the inverse of \( A \) from the right:

\[
X = B \cdot A^{-1}
\]

I will now compute the solutions for both parts using these approaches.Here are the solutions for the given problems:

### Part a: Solve \( A \cdot x = B \)
The solution for \( x \) is:
\[
x = \begin{pmatrix} 13 & 20 \\ -24 & -36 \end{pmatrix}
\]

### Part b: Solve \( X \cdot A = B \)
The solution for \( X \) is:
\[
X = \begin{pmatrix} -14 & 6 \\ 19 & -9 \end{pmatrix}
\]

Let me know if you'd like more detailed steps for these calculations or any further questions!

Here are five related questions:
1. How do you calculate the inverse of a matrix?
2. How do you know when a matrix has an inverse?
3. What is the geometric interpretation of matrix multiplication?
4. Can these solutions be verified by substituting back into the original equations?
5. How does matrix multiplication differ from scalar multiplication?

**Tip**: Always check if a matrix is invertible by computing its determinant. If the determinant is zero, the matrix is singular and has no inverse.

Solve the system of matrix equations A⋅x = B and X⋅A = B, where A and B are given 2x2 matrices.

Learn how to solve the matrix equations A⋅x = B and X⋅A = B, using matrix algebra and the inverse of A. This solution demonstrates step-by-step procedures, including inverse matrix calculations and matrix multiplication.

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