Question

If $B=\left[\begin{array}{ccc}5& 2\alpha & 1\\ 0& 2& 1\\ \alpha & 3& -1\end{array}\right]$is the inverse of a $3\times 3$ matrix $A$ then the sum of all values of $\alpha$ for which det $\left(A\right)+1=0$ is

Answer 1

Answer: 1

$\because B=A^{-1}=\Rightarrow \left|B\right|=\frac{1}{|A|}$
Now, $\left|B\right|=\left|\begin{array}{ccc}5& 2\alpha & 1\\ 0& 2& 1\\ \alpha & 3& -1\end{array}\right|=2{\alpha }^2-2\alpha -25$
Given, det $\left(A\right)+1=0$
$\Rightarrow \frac{1}{2{\alpha }^2-2\alpha -25}+1=0$
$\Rightarrow \frac{2{\alpha }^2-2\alpha -24}{2{\alpha }^2-2\alpha -25}=0$
$\Rightarrow \alpha =4-3$
$\Rightarrow$ sum of values of $\alpha =1$