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 $A^2$: