Question

If $\alpha$ is a characteristic root of a nonsingular matrix, then the corresponding characteristic root of adj A is

• A. $\frac{|A|}{\alpha }$
• B. $|\frac{A}{\alpha }|$
• C. $\frac{|adjA|}{\alpha }$
• D. $|\frac{adjA}{\alpha }|$

Answer 1

The correct option is A $\frac{|A|}{\alpha }$

Since $\alpha$ is a characteristic root of a nonsingular matrix,
therefore $\alpha \ne 0$. Also $\alpha$ is a characteristic root of
A implies that there exists a nonzero vector X such that
$AX=\alpha X$
$\left(adjA\right)\left(AX\right)=\left(adjA\right)\left(\alpha X\right)$
$\left[\right(adjA\left)A\right]X=\alpha \left(adjA\right)X$
$|A|IX=\alpha \left(adjA\right)X$ $[\because (adjA)A=|A|I]$
$|A|X=\alpha \left(adjA\right)X$
$|A|X=\alpha \left(adjA\right)X$
$\frac{|A|}{\alpha }X=\left(adjA\right)X$
$\left(adjA\right)X=\frac{|A|}{\alpha }X$
Since X is a nonzero vector, $|A/\alpha |$ is a characteristic root of the matrix adj A.