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We are given the matrix \( A = \begin{bmatrix} -2 & 1 \\ -5 & 3 \end{bmatrix} \) and are tasked to:

1. Evaluate \( A^2 - A \)
2. Express \( A^{-1} \) in terms of \( A \) and the identity matrix \( I \).

### Step 1: Compute \( A^2 \)

To compute \( A^2 \), we multiply \( A \) by itself:

\[
A^2 = A \cdot A = \begin{bmatrix} -2 & 1 \\ -5 & 3 \end{bmatrix} \cdot \begin{bmatrix} -2 & 1 \\ -5 & 3 \end{bmatrix}
\]

Performing matrix multiplication:

\[
A^2 = \begin{bmatrix} (-2)(-2) + (1)(-5) & (-2)(1) + (1)(3) \\ (-5)(-2) + (3)(-5) & (-5)(1) + (3)(3) \end{bmatrix}
\]

\[
A^2 = \begin{bmatrix} 4 - 5 & -2 + 3 \\ 10 - 15 & -5 + 9 \end{bmatrix} = \begin{bmatrix} -1 & 1 \\ -5 & 4 \end{bmatrix}
\]

### Step 2: Compute \( A^2 - A \)

Now, subtract \( A \) from \( A^2 \):

\[
A^2 - A = \begin{bmatrix} -1 & 1 \\ -5 & 4 \end{bmatrix} - \begin{bmatrix} -2 & 1 \\ -5 & 3 \end{bmatrix}
\]

\[
A^2 - A = \begin{bmatrix} -1 + 2 & 1 - 1 \\ -5 + 5 & 4 - 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
\]

Thus, \( A^2 - A = I \), the identity matrix.

### Step 3: Express \( A^{-1} \)

From the previous result, we have:

\[
A^2 - A = I
\]

This can be rewritten as:

\[
A(A - I) = 0
\]

Multiplying both sides by \( A^{-1} \), we get:

\[
A^{-1} = A - I
\]

Thus, the inverse of \( A \) is:

\[
A^{-1} = A - I = \begin{bmatrix} -2 & 1 \\ -5 & 3 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} -3 & 1 \\ -5 & 2 \end{bmatrix}
\]

### Summary:
- \( A^2 - A = I \)
- \( A^{-1} = A - I = \begin{bmatrix} -3 & 1 \\ -5 & 2 \end{bmatrix} \)

Would you like more details on any part of the solution?

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Here are some related questions:
1. What are the properties of the inverse of a matrix?
2. How can we verify if a matrix is invertible?
3. What other methods can be used to find the inverse of a matrix?
4. How does the determinant of a matrix relate to its inverse?
5. Can a non-square matrix have an inverse?

**Tip**: Always check if the determinant of a matrix is non-zero before attempting to find its inverse.

If A = [[-2, 1], [-5, 3]], evaluate A^2 - A. Hence express A^(-1) in terms of A and I.

Learn how to evaluate A^2 - A for matrix A = [[-2, 1], [-5, 3]] and express A^(-1) in terms of A and the identity matrix I. This solution covers key linear algebra concepts such as matrix multiplication, identity matrices, and matrix inversion.

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