If `A=[(3, 2, 0),( 1, 4 ,0 ),(0 ,0 ,5)]` , show that `A^2-7A+10 I_3=O` .

Explanation

To show that \( A^2 - 7A + 10I_3 = O \), where \( A = \begin{pmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{pmatrix} \) and \( I_3 \) is the identity matrix of order 3, we will follow these steps:

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

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

\[
A^2 = A \cdot A = \begin{pmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{pmatrix} \cdot \begin{pmatrix} 3 & 2 & 0 \\ 1 & 4 & 0 \\ 0 & 0 & 5 \end{pmatrix}
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