First slide

Applications of determinant

Question

Let M be a 3 x 3 non-singular matrix with
$\det (M) = α . If M^{- 1} adj (adj M) = k I, then the value of k is$

Moderate

A

1
B

$a$
C

$a^{2}$
D

$a^{3}$

Solution

We know that $M (a d j M) = d e t (M) I$
$\begin{array}{ll} \Rightarrow \det (M) M^{- 1} (adj (adj M)) = α^{2} I [∵ M^{- 1} = \frac{1}{| M |} adj (M)] \\ \Rightarrow α M^{- 1} (adj (adj M)) = α^{2} I \\ \Rightarrow M^{- 1} (adj (adj M)) = α I \\ But, M^{- 1} (adj (adj M)) = k \end{array}$
Hence, $k = α$

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