show that `[(1,1,3),(5,2,6),(-2,-1,-3)]=A` is nilpotent matrix of order 3.

Explanation

To show that the matrix \( A = \begin{pmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{pmatrix} \) is a nilpotent matrix of order 3, we need to demonstrate that \( A^3 = 0 \) (the zero matrix).

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

First, we need to find \( A^2 \) by multiplying matrix \( A \) by itself.

\[
A^2 = A \cdot A = \begin{pmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{pmatrix} \cdot \begin{pmatrix} 1 & 1 & 3 \\ 5 & 2 & 6 \\ -2 & -1 & -3 \end{pmatrix}
...