If `det(adjA)=|A|^(2),` then the order of matrix A is

Options

A. `2xx2`
B. `3xx3`
C. `2xx3`
D. `3xx2`

Answer

To solve the problem, we need to determine the order of the matrix \( A \) given that \( \text{det}(\text{adj} A) = |A|^2 \).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We know that the determinant of the adjoint of a matrix \( A \) is related to the determinant of \( A \) itself. The formula for the determinant of the adjoint of a matrix is:
\[ \text{det}(\text{adj} A) = |A|^{n-1} \]
where \( n \) is the order of the matrix \( A \).

2. **Setting Up the Equation**:
According to the problem, we have:
\[ \text{det}(\text{adj} A) = |A|^2 \]
From the formula, we can substitute:
\[ |A|^{n-1} = |A|^2 \]

3. **Comparing the Exponents**:
Since the determinants are equal, we can compare the exponents of \( |A| \):
\[ n - 1 = 2 \]

4. **Solving for \( n \)**:
To find \( n \), we solve the equation:
\[ n - 1 = 2 \implies n = 2 + 1 = 3 \]

5. **Conclusion**:
Therefore, the order of the matrix \( A \) is \( 3 \). This means that \( A \) is a \( 3 \times 3 \) matrix.

### Final Answer:
The order of matrix \( A \) is \( 3 \) (or \( 3 \times 3 \)).

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