Given that A is a square matrix of order `3 xx3` and |A| = - 4. Find |adj A|

Answer

To find the determinant of the adjoint of a square matrix \( A \) of order \( 3 \times 3 \) given that \( |A| = -4 \), we can use the following steps:

### Step 1: Understand the relationship between the determinant of a matrix and its adjoint.
The determinant of the adjoint of a matrix \( A \) can be calculated using the formula:
\[ |\text{adj } A| = |A|^{n-1} \]
where \( n \) is the order of the matrix. For a \( 3 \times 3 \) matrix, \( n = 3 \).

### Step 2: Substitute the known values into the formula.
We know that \( |A| = -4 \) and \( n = 3 \). Therefore, we can substitute these values into the formula:
\[ |\text{adj } A| = |A|^{3-1} = |A|^{2} \]

### Step 3: Calculate \( |\text{adj } A| \).
Now we can calculate \( |\text{adj } A| \):
\[ |\text{adj } A| = (-4)^{2} \]
Calculating this gives:
\[ |\text{adj } A| = 16 \]

### Final Answer:
Thus, the determinant of the adjoint of matrix \( A \) is:
\[ |\text{adj } A| = 16 \]

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