Question

If $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$ then find $Adj\left(AdjA\right)$.

• A. $\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$
• B. $\left[\begin{array}{ccc}3& 3& 4\\ 2& -3& -4\\ 0& -1& 1\end{array}\right]$
• C. $\left[\begin{array}{ccc}3& 3& 4\\ 2& -3& 4\\ 0& 1& 1\end{array}\right]$
• D. $\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& -4\\ 0& 1& 1\end{array}\right]$

Answer 1

The correct option is A $\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$

We know that $Adj\left(AdjA\right)=|A{|}^{n-2}A$
Since $A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$
Here $n=3$ (3 order matrix)
$\therefore |A|=\left|\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right|$
$=3\left(1\right)+3(2-0)+4(-2-0)$
$=1\ne 0,\therefore A$ is non-singular.
$\therefore Adj\left(AdjA\right)=|A{|}^{3-2}.A=|A|.A=1.A=A=\left[\begin{array}{ccc}3& -3& 4\\ 2& -3& 4\\ 0& -1& 1\end{array}\right]$