https://file.aiquickdraw.com/mathbot/file/17272265066611uezrlpc.jpg

Let me interpret and solve the problems from the image you've uploaded. It contains matrix operations and equation solving. I'll work through each of the visible questions.

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### Question 3: Solve the matrix equation.

Given:

\[
M = \begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}, 
S = \begin{pmatrix} a & c \\ b & d \end{pmatrix}, 
N = \begin{pmatrix} 4 & -8 \\ -3 & 2 \end{pmatrix}
\]

You need to find matrix \(S\) such that:

\[
M \cdot S = N
\]

This can be solved by multiplying \(M\) and \(S\) and equating the product to \(N\). 

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### Question 4: Solve for \(x\) and \(y\).

The system of equations is:

\[
\begin{aligned}
x - 2 & = 5 \\
y + 1 & = \frac{5}{-3}
\end{aligned}
\]

We can solve this system by directly solving for \(x\) and \(y\) in each equation.

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### Question 5: Determine the value of \((x_1^5 + x_2^5)\).

Given the determinant equation:

\[
\left| \begin{matrix} x - 2 & 1 \\ -1 & x \end{matrix} \right| = 2x - 4
\]

You need to calculate the determinant and find the roots \(x_1\) and \(x_2\). Then, compute \(x_1^5 + x_2^5\).

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I will now proceed with the detailed calculations.### Solution for Question 3:

The values of matrix \( S \) such that \( M \cdot S = N \) are:

\[
S = \begin{pmatrix} 15 & -26 \\ -26 & 44 \end{pmatrix}
\]

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I will now proceed to solve Question 4.### Solution for Question 4:

The solutions for the system of equations are:

\[
x = 7, \quad y = -\frac{8}{3}
\]

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Now, I will proceed to solve Question 5.### Solution for Question 5:

The roots of the determinant equation are:

\[
x_1 = 2 - i, \quad x_2 = 2 + i
\]

Now, calculating \(x_1^5 + x_2^5\), the result is:

\[
x_1^5 + x_2^5 = 244
\]

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Would you like further details on any of these solutions?

Here are 5 related questions to consider:
1. How do you compute the inverse of a matrix?
2. What are the properties of matrix multiplication?
3. How can you solve a system of linear equations using matrix operations?
4. What is the meaning of complex roots in the context of matrix determinants?
5. How would you extend this method to larger matrix sizes?

**Tip:** Always check the determinant of a matrix before finding its inverse to ensure it is not zero!

Solve the given matrix equations and systems of equations related to algebraic matrix operations.

This problem set involves solving matrix equations, finding the determinant of a matrix, and solving a system of equations. Key mathematical concepts include matrix multiplication, determinants, and the calculation of roots involving complex numbers. Ideal for students in Grades 10-12.

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