If 3, -2 are the Eigen values of a non-singular matrix A and |A| … - Vidyadip

MCQ

If $3, -2$ are the Eigen values of a non-singular matrix $A$ and $|A|\, = 4,$ then the Eigen values of $adj(A)$ are

A
$\frac{3}{4},\frac{{ - 1}}{2}$
✓
$\frac{4}{3}, - 2$
C
$12, -8$
D
$-12, 8$

✓

Answer

Correct option: B.

$\frac{4}{3}, - 2$

b
(b) Since ${A^{ - 1}} = \frac{{adj\,A}}{{|A|}}$ and if $\lambda$ is eigen value of $A$, then ${\lambda ^{ - 1}}$ is eigen value of ${A^{ - 1}}$.
Thus for $adj\,(A)X = ({A^{ - 1}}X)|A| = |A|{\lambda ^{ - 1}}I$.
Thus, eigen value corresponding to $\lambda = 3$ is $4/3$ and corresponding to $\lambda = - 2$ is $4/-2 = -2.$

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