Question

Statement-1 : If $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$, then $adj\left(adjA\right)=A$
Statement -2 : $|adj\left(adjA\right)|=|A{|}^{(n-1{)}^2}$, where A is a n-rowed non-singular square matrix

• A. Statement -1 is True, Statement-2 is True ; Statement-2 is a correct explanation for Statement-1
• B. Statement -1 is True, Statement-2 is True ; Statement-2 is not a correct explanation for Statement-1
• C. Statement-1 is True, Statement-2 is False
• D. Statement-1 is False, Statement-2 is True

Answer 1

The correct option is B Statement -1 is True, Statement-2 is True ; Statement-2 is not a correct explanation for Statement-1

$A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$
$\Rightarrow adjA={\left[\begin{array}{ccc}1& -2& -2\\ -1& 3& 3\\ 0& -4& -3\end{array}\right]}^T=\left[\begin{array}{ccc}1& -1& 0\\ -2& 3& -4\\ -2& 3& -3\end{array}\right]$
$adj\left(adjA\right)={\left[\begin{array}{ccc}3& 2& 0\\ -3& -3& -1\\ 4& 4& 1\end{array}\right]}^T=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]=A$
Hence, statement-1 is true.
Now, $|adj\left(adjA\right)|=|adjA{|}^{n-1}={\left(|A{|}^{(n-1)}\right)}^{n-1}=|A{|}^{(n-1{)}^2}$
Hence, statement 2 is true but does not explain statement 1