Matrices Question 3
Question 3 - 24 January - Shift 2
Let $A$ be a $3 \times 3$ matrix such that $|adj(adj(adj A))|=12^{4}$. Then $|A^{-1} adj A|$ is equal to
(1) $2 \sqrt{3}$
(2) $\sqrt{6}$
(3) 12
(4) 1
Show Answer
Answer: (1)
Solution:
Formula: Properties of Adjoint of a Matrix, Determinant properties
Given $|adj(adj(adj . A))|=12^{4}$
$\Rightarrow|A|^{(n-1)^{3}}=12^{4}$
Given $n=3$
$\Rightarrow|A|^{8}=12^{4}$
$\Rightarrow|A|^{2}=12$
$|A|=2 \sqrt{3}$
We are asked
$\mid A^{-1}$ adj $A \mid$
$=|A^{-1}| \cdot|adj A|$
$=\frac{1}{|A|} \cdot|A|^{3-1}$
$=|A|=2 \sqrt{3}$
