Question

For a \[3\times 3\] matrix A, if \[det\text{ }A\text{ }=\text{ }4\] , then \[\det \left( Adj.A \right)\] equals
(A) -4
(B) 4
(C) 16
(D) 64

Answer 1

Hint: The determinant value of matrix A is 4. Any matrix is invertible if and only if the determinant value of that matrix must not be equal to zero. So, matrix A is invertible. We know the property that the determinant value of the adjoint A is equal to the determinant of the matrix A to the power \[\left( n-1 \right)\] where A is an invertible \[n\times n\] square matrix. Use this property and get the determinant value of the adjoint of the matrix A.

Complete step-by-step answer:
According to the question, it is given that we have a \[3\times 3\] matrix A and the determinant value of matrix A is 4. We have to find the determinant value of the adjoint of the matrix A.
The determinant value of matrix A = 4 ……………………………(1)
We know the property that the determinant value of the adjoint A is equal to the determinant of the matrix A to the power \[\left( n-1 \right)\] where A is an invertible \[n\times n\] square matrix ………………………..(2)
Any matrix is invertible if and only if the determinant value of that matrix must not be equal to zero.
Since the determinant value of matrix A is equal to 4 so, matrix A is invertible.
It is also given that the matrix A is a \[3\times 3\] square matrix.
Now, we can use the property shown in equation (2) to get the determinant value of the adjoint of matrix A.
Using the property shown in equation (2), we get
\[\det \left( Adj.A \right)={{\left\{ \det \left( A \right) \right\}}^{3-1}}\] ………………………………(3)
From equation (1), we have the determinant value of matrix A.
Now, putting the determinant value of matrix A in equation (3), we get
 \[\begin{align}
  & \Rightarrow \det \left( Adj.A \right)={{\left\{ 4 \right\}}^{3-1}} \\
 & \Rightarrow \det \left( Adj.A \right)={{\left( 4 \right)}^{2}} \\
 & \Rightarrow \det \left( Adj.A \right)=16 \\
\end{align}\]
From the above equation, we have got the determinant value of the adjoint of matrix A.
Therefore, the determinant value of the adjoint of matrix A is 16.

Note: In this question, one might think to get the determinant of matrix A by getting the cofactors of the matrix A and the adjoint of matrix A. This approach is wrong because we don’t have sufficient information to get the cofactors and adjoint of the matrix A. So, don’t approach this question by this method.