If A is a 3-rowed square matrix and |A|=4 then |adjA|=?

Options

A. 4A
B. 16A
C. 64A
D. none of these

Explanation

To solve the problem, we need to find the determinant of the adjugate of a square matrix A, given that the determinant of A, denoted as |A|, is equal to 4.

### Step-by-Step Solution:

1. **Understanding the Adjugate Matrix**:
The adjugate (or adjoint) of a matrix A, denoted as adj A, is defined as the transpose of the cofactor matrix of A. For an n x n matrix A, the relationship between the determinant of A and the determinant of its adjugate is given by the formula:
\[ |adj A| = |A|^{n-1} \]
where n is the number of rows (or columns) in the square matrix A.

2. **Identifying the Size of the Matrix**:
In this case, A is a 3-rowed square matrix, which means it is a 3 x 3 matrix. Therefore, n = 3.

3. **Applying the Formula**:
We know that |A| = 4. Now we can substitute this value into the formula:
\[ |adj A| = |A|^{n-1} = |A|^{3-1} = |A|^{2} \]

4. **Calculating |adj A|**:
Now, substitute |A| = 4 into the equation:
\[ |adj A| = 4^{2} = 16 \]

5. **Final Result**:
Therefore, the determinant of the adjugate of matrix A is:
\[ |adj A| = 16 \]

### Summary:
The determinant of the adjugate of the matrix A, given that |A| = 4, is |adj A| = 16.