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

$α$

We know that $M (adj\, M) = det (M) I$

Replacing M by $adj\, M$, we get $adj\, M (adj (adj\, M)) = det (adj\, M) I$

$⇒det (M) M^{-1} (adj (adj\, M)) = α^2 I$  $[∵M^{-1}=\frac{1}{|M|}adj\, M]$

$⇒αM^{-1}(adj (adj\, M)) = α^2 I$

$⇒M^{-1}(adj (adj\, M)) = α I$

But, $M^{-1}(adj (adj\, M)) = k I$  [Given]

Hence, $k=α$.