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. $4$
• B. $11$
• C. $5$
• D. $0$

Answer 1

The correct option is C $5$

We have,

$P={\left|\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right|}_{(3\times 3)}$ and $|A|=4$

Now,

By Expansion Row

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

$=1(3\times 4-3\times 4)-\alpha (a-6)+3(1\times 4-2\times 3)$

$=1\times 0-\alpha (-1)+3(-2)$

$=2\alpha -6$

Given that

$|A|=4$

$2\alpha -6=4$

$2\alpha =4+6$

$2\alpha =10$

$\alpha =\frac{10}{2}$

$\alpha =5$

Hence, this is the answer.