Question

If P = $\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the adjoint of a 3x3 matrix A and det(A)=4,then α is equal to

• A. 4
• B. 11
• C. 5
• D. 7

Answer 1

The correct option is B 11

We have,
P = $\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$
First we will look at the determinants of the matrix P. The reason is that the determinant of a matrix and the determinant of it’s adjoint are related.
$\left|P\right|$ = 1(12-12) - $\alpha$ (4-6)+3(4-6) = 2$\alpha$ - 6
Given, P = adj(A)
Now the important expression required here is,
$\left|adjA\right|={\left|A\right|}^{n-1}$ (where n is the order of the matrix A).
$\left|P\right|=\left|adjA\right|={\left|A\right|}^2=16$ $(|adjA|=|A{|}^{n-1})$
$\Rightarrow 2\alpha -6=16$
$\Rightarrow \alpha =11$