If `A=[(1,0,0),(0,1,0),(0,0,1)]` then `A^(2)+2A` equals

Options

A. A
B. 2A
C. 3A
D. 4A

Answer

To solve the problem, we need to find \( A^2 + 2A \) where \( A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).

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

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

\[ A^2 = A \cdot A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

Calculating the elements of \( A^2 \):

- First row:
- First column: \( 1 \cdot 1 + 0 \cdot 0 + 0 \cdot 0 = 1 \)
- Second column: \( 1 \cdot 0 + 0 \cdot 1 + 0 \cdot 0 = 0 \)
- Third column: \( 1 \cdot 0 + 0 \cdot 0 + 0 \cdot 1 = 0 \)

- Second row:
- First column: \( 0 \cdot 1 + 1 \cdot 0 + 0 \cdot 0 = 0 \)
- Second column: \( 0 \cdot 0 + 1 \cdot 1 + 0 \cdot 0 = 1 \)
- Third column: \( 0 \cdot 0 + 1 \cdot 0 + 0 \cdot 1 = 0 \)

- Third row:
- First column: \( 0 \cdot 1 + 0 \cdot 0 + 1 \cdot 0 = 0 \)
- Second column: \( 0 \cdot 0 + 0 \cdot 1 + 1 \cdot 0 = 0 \)
- Third column: \( 0 \cdot 0 + 0 \cdot 0 + 1 \cdot 1 = 1 \)

Thus, we have:

\[ A^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

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

Next, we calculate \( 2A \):

\[ 2A = 2 \cdot \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix} \]

### Step 3: Add \( A^2 \) and \( 2A \)

Now, we add \( A^2 \) and \( 2A \):

\[ A^2 + 2A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} + \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix} \]

Calculating the elements:

- First row: \( 1 + 2 = 3 \), \( 0 + 0 = 0 \), \( 0 + 0 = 0 \)
- Second row: \( 0 + 0 = 0 \), \( 1 + 2 = 3 \), \( 0 + 0 = 0 \)
- Third row: \( 0 + 0 = 0 \), \( 0 + 0 = 0 \), \( 1 + 2 = 3 \)

Thus, we have:

\[ A^2 + 2A = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix} \]

### Final Result

The final result is:

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

where \( I \) is the identity matrix.