Let A be a square matrix of order 3 such that $adj. (adj. (adj. A)) =\begin{bmatrix}16&0&-24\\0&4&0\\0&12&4\end{bmatrix}$. Then find $|A|$

2

We know that $adj. (adj. A) = |A|^{n-2}A$, where n is order of matrix.

$∴adj. (adj. (adj. A)) = |adj. A|^{n-2} adj. A$

$=(|A|^{n-2})^{(n-2)} adj. A$

For n = 3,

$adj. (adj. (adj. A)) = |A|^2 adj. A =\begin{bmatrix}16&0&-24\\0&4&0\\0&12&4\end{bmatrix}$

$∴|A|^6|adj. A|=256$

$⇒|A|^6|A|^2=2^8$

$⇒|A|=2$