If A `=[{:(0,1,0),(0,0,1),(p,q,r):}]`, show that ltbargt `A^(3)= pI+qA+rA^(2)`

Explanation

To show that \( A^3 = pI + qA + rA^2 \) for the given matrix \( A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{pmatrix} \), we will first compute \( A^2 \) and then \( A^3 \).

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

To find \( A^2 \), we need to multiply matrix \( A \) by itself:

\[
A^2 = A \cdot A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ p & q & r \end{pmatrix}
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