Statement – 1 : If A is square matrix of order 3 such that $| A | = 3,$ then $| a d j (a d j (a d j A)) |$ is $3^{8}$
Statement – 2 : If A and B are square matrices of order $3 \times 3,$ where $| A | = 2$ and $| B | = 3,$ then $| (A^{- 1}) . a d j (B^{- 1}) . a d j (2 A^{- 1}) | = \frac{8}{9}$
Statement – 3 : A is a non-singular matrix of order $n \times n$ such that $3 A B A^{- 1} + A = 2 A^{- 1} B A$ then $| A + B | = 0$
The correct statements among the following are

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1,2&3

2,3

1,2

1,3

answer is D.

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Detailed Solution

Statement-1 : $| a d j (a d j A) | = {| A |}^{{(n - 1)}^{2}} | a d j (a d j (a d j (A))) | = {| A |}^{{(n - 1)}^{3} = 3^{8}}$

Statement-2 : $| A^{- 1} . a d j (B^{- 1}) . a d j (2 A^{- 1}) | = \frac{1}{| A |} . \frac{1}{{| B |}^{2}} . {| 2 A^{- 1} |}^{2}$ $= \frac{1}{| A |} . \frac{1}{{| B |}^{2}} \frac{64}{| A^{2} |} = \frac{64}{{| A |}^{3} {| B |}^{2}} = \frac{64}{8 \times 9} = \frac{8}{9}$