Let `A=[a_(ij)]_(nxxn)` be a square matrix and let `c_(ij)` be cofactor of `a_(ij)` in A. If `C=[c_(ij)]`, then

Options

A. |C|=|A|
B. `|C|=|A|^(n-1)`
C. `|C|=|A|^(n-2)`
D. None of these

Answer

To solve the problem, we need to establish the relationship between the determinant of the adjoint of matrix \( A \) and the determinant of the cofactor matrix \( C \). Let's go through the steps systematically:

### Step 1: Define the cofactor matrix
Let \( A = [a_{ij}]_{n \times n} \) be a square matrix. The cofactor \( c_{ij} \) of the element \( a_{ij} \) in matrix \( A \) is defined as:
\[ c_{ij} = (-1)^{i+j} \det(A_{ij}) \]
where \( A_{ij} \) is the submatrix obtained by deleting the \( i \)-th row and \( j \)-th column from \( A \).

### Step 2: Form the cofactor matrix
The cofactor matrix \( C \) is defined as:
\[ C = [c_{ij}]_{n \times n} \]
This matrix consists of all the cofactors of the elements of \( A \).

### Step 3: Relate the cofactor matrix to the adjoint
The adjoint (or adjugate) of matrix \( A \), denoted as \( \text{adj}(A) \), is the transpose of the cofactor matrix:
\[ \text{adj}(A) = C^T \]

### Step 4: Determine the determinant of the adjoint
The determinant of the adjoint of a matrix \( A \) can be expressed in terms of the determinant of \( A \):
\[ \det(\text{adj}(A)) = \det(C^T) = \det(C) \]
Since the determinant of a matrix is equal to the determinant of its transpose, we have:
\[ \det(C^T) = \det(C) \]

### Step 5: Use the property of determinants
For an \( n \times n \) matrix \( A \), the determinant of the adjoint can also be expressed as:
\[ \det(\text{adj}(A)) = \det(A)^{n-1} \]

### Step 6: Establish the final relationship
Combining the results from Step 4 and Step 5, we have:
\[ \det(C) = \det(\text{adj}(A)) = \det(A)^{n-1} \]

### Conclusion
Thus, we conclude that:
\[ \det(C) = \det(A)^{n-1} \]