QJEEAdv20Ma7c07: JEE Advanced 2020 Mathematics - Class 12 JEEAdv20Ma7c07 Solution: Let M be a 3 3 invertible matrix with real entries and let I…

It is given that matrix M be a 3 $\times$ 3 invertible matrix, such that

M ^{$-$ 1} = adj(adj M) $\Rightarrow$ M ^{$-$ 1} = |M| M

( $\because$ for a matrix A of order 'n' adj(adjA) = |A|^{n $-$ 2} A}

$\Rightarrow$ M ^{$-$ 1} M = |M|M² $\Rightarrow$ M²|M| = I .....(i)

$\because$ det(M² |M|) = det(I) = 1

$\Rightarrow$ |M|³|M|² = 1 $\Rightarrow$ |M| = 1 .....(ii)

from Eqs. (i) and (ii), we get

M² = I

As, adj M = |M|M ^{$-$ 1} = M

$\Rightarrow$ (adj M)² = M² (adj M)² = I