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The image contains five math problems involving matrices, determinants, and systems of equations. Let’s solve these problems one by one:

### Problem 1:
Matrix \( A = \begin{pmatrix} 4 & -1 \\ 2 & 5 \end{pmatrix} \) and matrix \( B = \begin{pmatrix} 2 & -1 \\ 3 & -4 \end{pmatrix} \). 

We are tasked with finding \( (BA)(B + A^{-1})B^{-1} \).

### Solution Steps:
1. **Compute the product \( BA \):**
   \[
   BA = \begin{pmatrix} 2 & -1 \\ 3 & -4 \end{pmatrix} \begin{pmatrix} 4 & -1 \\ 2 & 5 \end{pmatrix}
   \]
2. **Find the inverse of matrix \( A \), denoted as \( A^{-1} \).**
   \[
   A^{-1} = \frac{1}{\text{det}(A)} \text{adj}(A)
   \]
   First, calculate the determinant of \( A \), \( \det(A) = (4)(5) - (2)(-1) = 20 + 2 = 22 \).

3. **Then, calculate \( B + A^{-1} \), and finally the expression.**

Would you like me to fully calculate this, or would you prefer to move to the next question first?

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### Problem 2:
Matrix \( A = \begin{pmatrix} 5 & -2 \\ -3 & 6 \end{pmatrix} \). We are tasked to find the determinant of \( (A^2)^T \).

### Solution Steps:
1. **First, compute \( A^2 \),** which is \( A \times A \).
2. **Transpose \( A^2 \) to get \( (A^2)^T \).**
3. **Find the determinant of \( (A^2)^T \).**

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### Problem 3:
Matrix \( P = \begin{pmatrix} -3 & -6 & 2 \\ 1 & 3 & 1 \\ 3 & 4 & 5 \end{pmatrix} \). The goal is to find the value of \( P^T P^{-1} \).

### Solution Steps:
1. **Transpose \( P \) to get \( P^T \).**
2. **Find the inverse of matrix \( P \).**
3. **Multiply \( P^T \) and \( P^{-1} \).**

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### Problem 4:
We are given the equation involving a determinant:
\[
\left| \begin{pmatrix} x - 3 & x - 1 \\ 2x - 3 & 1 \end{pmatrix} \right| = 0
\]
Find \( x_1^2 + x_2^2 \), where \( x_1 \) and \( x_2 \) are the roots.

### Solution Steps:
1. **Calculate the determinant and solve for \( x_1 \) and \( x_2 \).**
2. **Sum their squares.**

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### Problem 5:
We are given a system of linear equations:
\[
\begin{pmatrix} 1 & 3 & 0 \\ 2 & 1 & 4 \\ 2 & -4 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 4 \\ 6 \\ -8 \end{pmatrix}
\]
Find \( 3x - 2y + z \).

### Solution Steps:
1. **Solve the system of equations using methods like Gaussian elimination or matrix inversion.**
2. **Find \( 3x - 2y + z \).**

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Would you like me to go through a full solution for each problem, or should I focus on a specific one? Let me know!

### Related Questions:
1. How do you compute the inverse of a matrix manually?
2. What is the significance of the determinant in matrix theory?
3. How do you solve a system of linear equations using matrices?
4. What is matrix transposition, and why is it important?
5. Can the determinant of a matrix be zero? What does that imply?

### Tip:
When dealing with complex matrix operations, break down the steps (multiplication, inversion, etc.) to ensure accuracy in the intermediate calculations.

The image contains five problems related to matrices, determinants, and systems of equations.

Explore a step-by-step guide on solving complex matrix operations including inverse matrices, determinants, and systems of linear equations. These problems require a solid understanding of linear algebra concepts. Suitable for Grades 11-12.

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