https://file.aiquickdraw.com/mathbot/file/1726057773547lc25z0n9.jpg

You uploaded a set of matrix-related problems. I can solve them step by step for you. Let's start with **problem 6** from the image, which involves finding the matrix \( X \) that will solve the given matrix equations.

### Problem 6(a):
Solve for \( X \) in:
\[
5X - \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix} = 3X - \begin{bmatrix} 5 & 2 \\ 6 & 1 \end{bmatrix}
\]

We can rearrange the equation as:
\[
5X - 3X = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix} - \begin{bmatrix} 5 & 2 \\ 6 & 1 \end{bmatrix}
\]
\[
2X = \begin{bmatrix} 1 - 5 & 0 - 2 \\ 2 - 6 & 3 - 1 \end{bmatrix}
\]
\[
2X = \begin{bmatrix} -4 & -2 \\ -4 & 2 \end{bmatrix}
\]

Dividing both sides by 2:
\[
X = \begin{bmatrix} -2 & -1 \\ -2 & 1 \end{bmatrix}
\]

### Problem 6(b):
Solve for \( X \) in:
\[
\begin{bmatrix} 2 & 1 \\ 2 & 2 \end{bmatrix} X = I_2
\]

This is a matrix equation where \( I_2 \) is the identity matrix. We need to find the inverse of the matrix on the left-hand side, as multiplying by the inverse will give \( X \).

Let me calculate the inverse of the matrix and solve for \( X \).The inverse of the matrix \( \begin{bmatrix} 2 & 1 \\ 2 & 2 \end{bmatrix} \) is:
\[
X = \begin{bmatrix} 1 & -0.5 \\ -1 & 1 \end{bmatrix}
\]

### Problem 6(c):
Solve for \( X \) in:
\[
\begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix} X - X \begin{bmatrix} 1 & 4 \\ 2 & 0 \end{bmatrix} = \begin{bmatrix} 2 & -2 \\ 5 & 4 \end{bmatrix}
\]

This equation can be solved by combining like terms and solving for \( X \). Let me handle that next.

Would you like me to proceed with solving part 6(c) or jump to another part?

Solve the matrix equations provided in the image for matrix X in different parts of problem 6.

Learn how to solve matrix equations to find the matrix X, using linear algebra concepts such as matrix inversion and multiplication. Step-by-step solutions are provided for different parts of the problem, covering the properties and operations of matrices. Suitable for undergraduate students.

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