Question

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

• A. $4$
• B. $11$
• C. $5$
• D. $0$

Answer 1

The correct option is B $11$

$\left|\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right|=1(3\times 4-3\times 4)-\alpha (1\times 4-3\times 2)+3(1\times 4-3\times 2)=2\alpha -6$

$\left|\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right|=\left|\begin{array}{c}adj.A\end{array}\right|$

$\Rightarrow (2\alpha -6)=\left|\begin{array}{c}adj.A\end{array}\right|$

Now,

$\left|\begin{array}{c}adj.A\end{array}\right|={\left|A\right|}^{3-1}=4^2=16$

$\therefore 2\alpha -6=16$

$\Rightarrow 2\alpha =16+6$

$\Rightarrow \alpha =\frac{22}{2}=11$