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?

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 $(A^2)^T$.

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 $P^{-1}$.

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.

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 $3x - 2y + z$.

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.