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Algebraic Identities

Let M be a ...

Question

Let $M$ be a $3 \times 3$ non-singular matrix with $d e t (M) = α$ . If $[M^{- 1} a d j (a d j (M)] = K I$ , then the value of $K$ is

A

1

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B

α

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C

α^{2}

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D

α^{3}

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Solution

The correct option is C $α$
Formulae:-
$A^{- 1} = \frac{1}{| A |} a d j (A)$
$A \cdot a d j (A) = | A | I = a d j (A) \cdot A$
$a d j (a d j A) = {| A |}^{(n - 2)} A$
Now,
$[M^{- 1} a d j (a d j (M)]$
$= \frac{1}{| M |} a d j (M) \cdot | M |^{(3 - 2)} M$
$= a d j (M) \cdot M$
$= | M | I$
$= α I$
$∴ k = α$

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