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The problem asks to show that for the given matrix \( A \), the equation \( A^2 - 2A - 5I = 0 \) holds true. The matrix \( A \) is:

\[
A = \begin{pmatrix} -1 & 2 \\ 1 & 3 \end{pmatrix}
\]

Where \( I \) is the identity matrix, and we are tasked with proving the equation involving matrix powers and operations.

### Steps:

1. **Calculate \( A^2 \):**
   \[
   A^2 = A \times A = \begin{pmatrix} -1 & 2 \\ 1 & 3 \end{pmatrix} \times \begin{pmatrix} -1 & 2 \\ 1 & 3 \end{pmatrix}
   \]
   
   Perform the matrix multiplication.

2. **Multiply matrix \( A \) by 2:**
   \[
   2A = 2 \times \begin{pmatrix} -1 & 2 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} -2 & 4 \\ 2 & 6 \end{pmatrix}
   \]

3. **Multiply identity matrix \( I \) by 5:**
   \[
   5I = 5 \times \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}
   \]

4. **Substitute these values into the equation \( A^2 - 2A - 5I \) and verify if the result is the zero matrix:**

\[
A^2 - 2A - 5I = ?
\]

Let's calculate these steps in detail:

#### 1. Calculating \( A^2 \):
\[
A^2 = \begin{pmatrix} -1 & 2 \\ 1 & 3 \end{pmatrix} \times \begin{pmatrix} -1 & 2 \\ 1 & 3 \end{pmatrix}
= \begin{pmatrix} (-1)(-1) + (2)(1) & (-1)(2) + (2)(3) \\ (1)(-1) + (3)(1) & (1)(2) + (3)(3) \end{pmatrix}
= \begin{pmatrix} 1 + 2 & -2 + 6 \\ -1 + 3 & 2 + 9 \end{pmatrix}
= \begin{pmatrix} 3 & 4 \\ 2 & 11 \end{pmatrix}
\]

#### 2. Now calculate \( A^2 - 2A - 5I \):
\[
A^2 - 2A = \begin{pmatrix} 3 & 4 \\ 2 & 11 \end{pmatrix} - \begin{pmatrix} -2 & 4 \\ 2 & 6 \end{pmatrix}
= \begin{pmatrix} 3 - (-2) & 4 - 4 \\ 2 - 2 & 11 - 6 \end{pmatrix}
= \begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}
\]

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

Thus, we have proven that \( A^2 - 2A - 5I = 0 \).

Would you like further clarification on any step? Here are some related questions:

1. What is the general process for matrix multiplication?
2. How do we prove similar equations with larger matrices?
3. What is the significance of the identity matrix in such problems?
4. How would we find the inverse of matrix \( A \)?
5. Can this equation be generalized for other \( n \times n \) matrices?

**Tip:** Matrix multiplication is not commutative, so the order of multiplication matters.

Tunjukkan bahwa A^2 - 2A - 5I = 0 untuk matriks A = [[-1, 2], [1, 3]].

Learn how to prove the equation A^2 - 2A - 5I = 0 for the given matrix A = [[-1, 2], [1, 3]]. This problem explores key linear algebra concepts, including matrix multiplication, scalar multiplication, and identity matrix properties.

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