If $A=\begin{bmatrix}3&0&0\ 0&3&0\ 0&0&3\end{bmatrix}$, then $A^{5}=$

Q: If $A=\begin{bmatrix}3&0&0\\ 0&3&0\\ 0&0&3\end{bmatrix}$, then $A^{5}=$

If $A=\begin{bmatrix}3&0&0\\ 0&3&0\\ 0&0&3\end{bmatrix}$, then $A^{5}=$

Question

Accepted Answer

To determine $ A^5 $ for the matrix $ A = \begin{bmatrix} 3 & 0 & 0 \ 0 & 3 & 0 \ 0 & 0 & 3 \end{bmatrix} $, we need to understand the properties of diagonal matrices and matrix exponentiation.

### Explanation:

1. **Properties of Diagonal Matrices:**
   - A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The matrix $ A $ given here is a diagonal matrix with all diagonal elements equal to 3.
   - When raising a diagonal matrix to a power, you simply raise each of the diagonal elements to that power. This is because the off-diagonal elements remain zero, and the diagonal elements are unaffected by the multiplication of zeros.

2. **Calculating $ A^5 $:**
   - Since $ A $ is a diagonal matrix, $ A^5 $ is also a diagonal matrix where each diagonal element is raised to the power of 5.
   - Therefore, $ A^5 = \begin{bmatrix} 3^5 & 0 & 0 \ 0 & 3^5 & 0 \ 0 & 0 & 3^5 \end{bmatrix} $.

3. **Computing $ 3^5 $:**
   - $ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 $.

4. **Resulting Matrix:**
   - Thus, $ A^5 = \begin{bmatrix} 243 & 0 & 0 \ 0 & 243 & 0 \ 0 & 0 & 243 \end{bmatrix} $.
   - This matrix can be expressed as $ 243 \times A $, since multiplying the original matrix $ A $ by 243 scales each of its diagonal elements by 243.

### Justification for Option 1:

- **Option 1: 243 A** is correct because $ A^5 = 243 \times A $, as shown by the calculation above.

### Explanation for Incorrect Options:

- **Option 2: 81 A** is incorrect because $ 3^4 = 81 $, not $ 3^5 $. This option mistakenly uses the fourth power instead of the fifth.
  
- **Option 3: 27 A** is incorrect because $ 3^3 = 27 $, not $ 3^5 $. This option mistakenly uses the third power instead of the fifth.

- **Option 4: 9 A** is incorrect because $ 3^2 = 9 $, not $ 3^5 $. This option mistakenly uses the second power instead of the fifth.

In conclusion, the correct answer is Option 1: 243 A, as it accurately represents the result of raising the matrix $ A $ to the fifth power.