Question

If $P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the adjoint of a $3\times 3$ matrix $A$ and $|A|=4,$ then $\alpha$ is equal to

• A. 11
• B. 5
• C. 0
• D. 4

Answer 1

Answer: (A)

$P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$
Let $|P|=1(12-12)-\alpha (4-6)+3(4-6)=2\alpha -6$
Also, det(adj A) = (det A)${}^2$.
$\Rightarrow 2\alpha -6=16\Rightarrow 2\alpha =22.\therefore \alpha =11$
Remark: det(adj $A$):= (det $A$)${}^{n-1}$· where A is a matrix of order n.