Let A be a non-singular square matrix of order `3xx3`. Then `|adjA|` is equal to

Options

A. `|A|`
B. `|A|^(2)`
C. `|A|^(3)`
D. `3|A|`

Answer

To find the value of \(|\text{adj} A|\) for a non-singular square matrix \(A\) of order \(3 \times 3\), we can follow these steps:

### Step 1: Understand the relationship between the determinant of a matrix and its adjugate.
The determinant of the adjugate of a matrix \(A\) is given by the formula:
\[ |\text{adj} A| = |A|^{n-1} \]
where \(n\) is the order of the matrix.

### Step 2: Identify the order of the matrix.
In this case, the order of matrix \(A\) is \(3\) (since it is a \(3 \times 3\) matrix).

### Step 3: Substitute the order into the formula.
Using the formula from Step 1, we substitute \(n = 3\):
\[ |\text{adj} A| = |A|^{3-1} = |A|^{2} \]

### Step 4: Determine the value of \(|A|\).
Since \(A\) is a non-singular matrix, we know that \(|A| \neq 0\).

### Step 5: Conclude the result.
Thus, we have:
\[ |\text{adj} A| = |A|^{2} \]

### Final Answer:
The value of \(|\text{adj} A|\) is \(|A|^{2}\).

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