The image you've provided shows a matrix equation involving the matrix $A$ and a request to find $A^2 - 5A + 4I$, where $I$ is the identity matrix.

Let's first extract the matrix $A$:

$A = \begin{bmatrix} -2 & 1 & 0 \\ -1 & 4 & 3 \\ 0 & 8 & 5 \end{bmatrix}$

Now, we are asked to find the expression $A^2 - 5A + 4I$. To proceed, we will:

1. Compute $A^2$ (the square of the matrix $A$),
2. Compute $5A$ (five times matrix $A$),
3. Compute $4I$ (four times the identity matrix),
4. Combine these terms to find $A^2 - 5A + 4I$.

Let me perform these calculations for you. numpy as np

Defining the matrix A

A = np.array([[-2, 1, 0], [-1, 4, 3], [0, 8, 5]])

Identity matrix of the same size as A

I = np.identity(3)

Calculating A^2, 5A, and 4I

A_squared = np.dot(A, A) five_A = 5 * A four_I = 4 * I

Finding A^2 - 5A + 4I

result = A_squared - five_A + four_I resultThe result of the expression $A^2 - 5A + 4I$ is:

$A^2 - 5A + 4I = \begin{bmatrix} 17 & -3 & 3 \\ 3 & 23 & 12 \\ -8 & 32 & 28 \end{bmatrix}$

Would you like any further details or explanations about the steps?

Here are some related questions for further exploration:

1. How do you compute the square of a matrix $A$?
2. What is the role of the identity matrix in matrix operations?
3. How does matrix multiplication differ from scalar multiplication?
4. Can this process be applied to non-square matrices?
5. What other types of matrix equations are commonly used in linear algebra?

Tip: When multiplying matrices, always check if the number of columns in the first matrix matches the number of rows in the second matrix.