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
Answer & explanation
Correct answer: option 2
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, $k=α$.