If `A=[{:(2,0),(0,2):}]` then `A^(2) - 3I`=……

Options

A. O
B. I
C. A
D. 3A

Answer

To solve the problem, we need to calculate \( A^2 - 3I \) where \( A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \) and \( I \) is the identity matrix.

### Step-by-Step Solution:

**Step 1: Calculate \( A^2 \)**
To find \( A^2 \), we multiply matrix \( A \) by itself.

\[ A^2 = A \cdot A = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \cdot \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \]

Calculating the product:

\[ A^2 = \begin{pmatrix} (2 \cdot 2 + 0 \cdot 0) & (2 \cdot 0 + 0 \cdot 2) \\ (0 \cdot 2 + 2 \cdot 0) & (0 \cdot 0 + 2 \cdot 2) \end{pmatrix} = \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} \]

**Step 2: Calculate \( 3I \)**
The identity matrix \( I \) for a \( 2 \times 2 \) matrix is:

\[ I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

Now, we multiply \( I \) by 3:

\[ 3I = 3 \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \]

**Step 3: Calculate \( A^2 - 3I \)**
Now we subtract \( 3I \) from \( A^2 \):

\[ A^2 - 3I = \begin{pmatrix} 4 & 0 \\ 0 & 4 \end{pmatrix} - \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \]

Calculating the subtraction:

\[ A^2 - 3I = \begin{pmatrix} (4 - 3) & (0 - 0) \\ (0 - 0) & (4 - 3) \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \]

**Final Result:**

\[ A^2 - 3I = I \]