If A is a non - null diagonal matrix of order 3 such that `A^(4)=A^(2)`, then the possible number of matrices A are

Options

A. 27
B. 26
C. 8
D. 7

Answer

To solve the problem, we need to analyze the given conditions for the diagonal matrix \( A \) of order 3, such that \( A^4 = A^2 \).

### Step-by-Step Solution:

1. **Understanding the Matrix**:
Since \( A \) is a diagonal matrix of order 3, we can represent it as:
\[ A = \begin{pmatrix} d_1 & 0 & 0 \\ 0 & d_2 & 0 \\ 0 & 0 & d_3 \end{pmatrix} \]
where \( d_1, d_2, d_3 \) are the diagonal elements.

2. **Calculating \( A^2 \) and \( A^4 \)**:
- The square of \( A \) is:
\[ A^2 = \begin{pmatrix} d_1^2 & 0 & 0 \\ 0 & d_2^2 & 0 \\ 0 & 0 & d_3^2 \end{pmatrix} \]
- The fourth power of \( A \) is:
\[ A^4 = \begin{pmatrix} d_1^4 & 0 & 0 \\ 0 & d_2^4 & 0 \\ 0 & 0 & d_3^4 \end{pmatrix} \]

3. **Setting Up the Equation**:
According to the problem, we have:
\[ A^4 = A^2 \]
This implies:
\[ \begin{pmatrix} d_1^4 & 0 & 0 \\ 0 & d_2^4 & 0 \\ 0 & 0 & d_3^4 \end{pmatrix} = \begin{pmatrix} d_1^2 & 0 & 0 \\ 0 & d_2^2 & 0 \\ 0 & 0 & d_3^2 \end{pmatrix} \]

4. **Equating the Diagonal Elements**:
From the equality of the matrices, we get the following equations:
- \( d_1^4 = d_1^2 \)
- \( d_2^4 = d_2^2 \)
- \( d_3^4 = d_3^2 \)

5. **Solving the Equations**:
Each equation can be factored as:
\[ d_i^4 - d_i^2 = 0 \implies d_i^2(d_i^2 - 1) = 0 \]
This gives us the solutions:
- \( d_i^2 = 0 \) (i.e., \( d_i = 0 \))
- \( d_i^2 = 1 \) (i.e., \( d_i = 1 \) or \( d_i = -1 \))

Therefore, for each \( d_i \) (where \( i = 1, 2, 3 \)), the possible values are:
- \( d_i = 0 \)
- \( d_i = 1 \)
- \( d_i = -1 \)

6. **Counting the Possibilities**:
Since \( A \) is a non-null matrix, at least one of \( d_1, d_2, d_3 \) must be non-zero. Each diagonal element has 3 choices (0, 1, -1). Thus, the total combinations without restriction would be:
\[ 3^3 = 27 \]
However, we need to exclude the case where all \( d_1, d_2, d_3 \) are 0 (which gives the null matrix). There is only 1 such case.

Therefore, the total number of valid matrices \( A \) is:
\[ 27 - 1 = 26 \]

### Final Answer:
The possible number of matrices \( A \) is **26**.