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### Problem 3:
You are given three matrices:
- \( M = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix} \)
- \( A = \begin{pmatrix} -8 & -3 \\ 4 & 1 \end{pmatrix} \)
- \( N = \begin{pmatrix} 4 & -3 \\ -3 & 2 \end{pmatrix} \)

You are asked to find matrix \( S = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) such that it satisfies two matrix equations:
- \( M \cdot A \cdot S = N \)
- \( M \cdot S \cdot A = N \)

### Problem 4:
Solve for all values of \( x \) and \( y \) that are solutions to the matrix equation:
\[
\begin{pmatrix} x - 2 & 5 \\ -3 & y + 1 \end{pmatrix} \cdot \begin{pmatrix} x - 2 & 5 \\ -3 & y + 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]
This is a matrix equation, and we are asked to find \( x \) and \( y \) that make the product of the matrices equal to the identity matrix.

### Problem 5:
You are asked to compute \( x_1^5 + x_2^5 \) given the roots \( x_1 \) and \( x_2 \) of the determinant equation:
\[
\begin{vmatrix} x - 2 & 1 & 2x - 4 \\ -1 & x - 4 & -1 \\ -1 & -2 & 2x - 3 \end{vmatrix} = 0
\]
This is a determinant equation where you need to solve for the roots \( x_1 \) and \( x_2 \), and then compute the fifth power of each root and sum them.

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Here are 5 questions you might consider next:
1. How do we approach solving a system of matrix equations like in problem 3?
2. What properties of matrix multiplication are useful in solving problem 4?
3. How do we calculate determinants of 3x3 matrices to solve problem 5?
4. What methods can be used to solve for the unknowns \( x_1 \) and \( x_2 \) in problem 5?
5. How does matrix inversion help in finding solutions for \( x \) and \( y \)?

**Tip:** For solving matrix equations, knowing how to multiply matrices and use inverses is essential.

Solve the given mathematical problems related to matrices and determinant equations. The questions involve matrix multiplication, finding specific values of variables that satisfy matrix equations, and computing powers of roots obtained from a determinant equation.

Explore step-by-step solutions for matrix multiplication equations and determinant calculations. This problem set covers advanced algebraic concepts, including matrix multiplication, identity matrix properties, and solving determinant equations. Ideal for high school students in Grades 10-12.

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