Let A, B, C, D be non singular matrices of order $3 \times 3$ such that $(A A^{T}) B C^{2} = B^{T}$ and $C^{2} = I$ and D is an orthogonal matrix such that $A^{T}$ and D commute with each other and $D^{T} A^{T} D = A^{- 1}$ , then
Column-I
Column-II
A)
If $B = B^{- 1}$ , then $| B^{2018} C |$ can be equal to
P)
–1
B)
If ${(A C A^{T})}^{2019} = A C^{K} A^{T}$ then k can be equal to
Q)
0
C)
$| A C A^{T} |$ can be equal to
R)
2
D)
$| B B^{T} {(B^{- 1})}^{2} | + | {(A A^{T})}^{2018} |$ is equal to
S)
3

T)
4

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$A - P, B - S, C - R, D - R$

$A - P, B - R, C - S, D - P$

$A - S, B - P, C - R, D - P$

$A - P, B - S, C - P, D - R$

answer is B.

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Detailed Solution

$A^{T} D = D A^{T}$
$D D^{T} = D^{T} D = I$
$\Rightarrow D D^{T} A^{T} D = D A^{T} D A^{1} = D A^{T} \Rightarrow A^{- 1} = A^{T}$
$\Rightarrow B C^{2} = B^{T} B = B^{T}$
(A) $B = B^{- 1} \Rightarrow B^{2} = I$
$\Rightarrow | B^{2018} C | = | C | = \pm 1$
(B) ${(A C A^{T})}^{2019} = (A C A^{T}) (A C A^{T}) ....... (A C T^{T}) = A C^{2019} A^{T} = A C^{k} A^{T} \Rightarrow k$
is odd
(C) $| A C T^{T} | = {| A |}^{2} | C | = | C | = \pm 1$
(D) $| B B^{T} {(B^{- 1})}^{2} | + | {(A A^{T})}^{2018} | = 1 + 1 = 2$

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Column-I			Column-II
A)	If $B = B^{- 1}$ , then $\| B^{2018} C \|$ can be equal to	P)		–1
B)	If ${(A C A^{T})}^{2019} = A C^{K} A^{T}$ then k can be equal to	Q)		0
C)	$\| A C A^{T} \|$ can be equal to	R)		2
D)	$\| B B^{T} {(B^{- 1})}^{2} \| + \| {(A A^{T})}^{2018} \|$ is equal to	S)		3
		T)		4