If `A=[[2, 0, 0], [0, 2, 0], [0, 0, 2]]`, then `A^(5)=`

Options

A. `32A`
B. `16A`
C. `10A`
D. `5A`

Answer

To find \( A^5 \) for the matrix

\[ A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]

we can follow these steps:

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

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

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

Calculating the product:

- First row:
- \( 2 \cdot 2 + 0 \cdot 0 + 0 \cdot 0 = 4 \)
- \( 2 \cdot 0 + 0 \cdot 2 + 0 \cdot 0 = 0 \)
- \( 2 \cdot 0 + 0 \cdot 0 + 0 \cdot 2 = 0 \)

- Second row:
- \( 0 \cdot 2 + 2 \cdot 0 + 0 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 2 \cdot 2 + 0 \cdot 0 = 4 \)
- \( 0 \cdot 0 + 2 \cdot 0 + 0 \cdot 2 = 0 \)

- Third row:
- \( 0 \cdot 2 + 0 \cdot 0 + 2 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 0 \cdot 2 + 2 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 0 \cdot 0 + 2 \cdot 2 = 4 \)

Thus,

\[ A^2 = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix} \]

### Step 2: Calculate \( A^3 \)

Now, we calculate \( A^3 \) by multiplying \( A^2 \) by \( A \):

\[ A^3 = A^2 \cdot A = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{bmatrix} \cdot \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]

Calculating the product:

- First row:
- \( 4 \cdot 2 + 0 \cdot 0 + 0 \cdot 0 = 8 \)
- \( 4 \cdot 0 + 0 \cdot 2 + 0 \cdot 0 = 0 \)
- \( 4 \cdot 0 + 0 \cdot 0 + 0 \cdot 2 = 0 \)

- Second row:
- \( 0 \cdot 2 + 4 \cdot 0 + 0 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 4 \cdot 2 + 0 \cdot 0 = 8 \)
- \( 0 \cdot 0 + 4 \cdot 0 + 0 \cdot 2 = 0 \)

- Third row:
- \( 0 \cdot 2 + 0 \cdot 0 + 4 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 0 \cdot 2 + 4 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 0 \cdot 0 + 4 \cdot 2 = 8 \)

Thus,

\[ A^3 = \begin{bmatrix} 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{bmatrix} \]

### Step 3: Calculate \( A^4 \)

Next, we calculate \( A^4 \):

\[ A^4 = A^3 \cdot A = \begin{bmatrix} 8 & 0 & 0 \\ 0 & 8 & 0 \\ 0 & 0 & 8 \end{bmatrix} \cdot \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]

Calculating the product:

- First row:
- \( 8 \cdot 2 + 0 \cdot 0 + 0 \cdot 0 = 16 \)
- \( 8 \cdot 0 + 0 \cdot 2 + 0 \cdot 0 = 0 \)
- \( 8 \cdot 0 + 0 \cdot 0 + 0 \cdot 2 = 0 \)

- Second row:
- \( 0 \cdot 2 + 8 \cdot 0 + 0 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 8 \cdot 2 + 0 \cdot 0 = 16 \)
- \( 0 \cdot 0 + 8 \cdot 0 + 0 \cdot 2 = 0 \)

- Third row:
- \( 0 \cdot 2 + 0 \cdot 0 + 8 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 0 \cdot 2 + 8 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 0 \cdot 0 + 8 \cdot 2 = 16 \)

Thus,

\[ A^4 = \begin{bmatrix} 16 & 0 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 16 \end{bmatrix} \]

### Step 4: Calculate \( A^5 \)

Finally, we calculate \( A^5 \):

\[ A^5 = A^4 \cdot A = \begin{bmatrix} 16 & 0 & 0 \\ 0 & 16 & 0 \\ 0 & 0 & 16 \end{bmatrix} \cdot \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]

Calculating the product:

- First row:
- \( 16 \cdot 2 + 0 \cdot 0 + 0 \cdot 0 = 32 \)
- \( 16 \cdot 0 + 0 \cdot 2 + 0 \cdot 0 = 0 \)
- \( 16 \cdot 0 + 0 \cdot 0 + 0 \cdot 2 = 0 \)

- Second row:
- \( 0 \cdot 2 + 16 \cdot 0 + 0 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 16 \cdot 2 + 0 \cdot 0 = 32 \)
- \( 0 \cdot 0 + 16 \cdot 0 + 0 \cdot 2 = 0 \)

- Third row:
- \( 0 \cdot 2 + 0 \cdot 0 + 16 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 0 \cdot 2 + 16 \cdot 0 = 0 \)
- \( 0 \cdot 0 + 0 \cdot 0 + 16 \cdot 2 = 32 \)

Thus,

\[ A^5 = \begin{bmatrix} 32 & 0 & 0 \\ 0 & 32 & 0 \\ 0 & 0 & 32 \end{bmatrix} \]

### Final Answer

Therefore,

\[ A^5 = 32 \cdot I = 32 \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 32 & 0 & 0 \\ 0 & 32 & 0 \\ 0 & 0 & 32 \end{bmatrix} \]