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JEE Main · 2014 · 06 Apr 2014 · Offline · Matrices

If A is an

3 \times 3

non-singular matrix such that

A A^{'} = A^{'} A

and

B = A^{- 1} A^{'}

, then

B B^{'}

equals

(A) $I + B$
(B)I✓ Correct
(C) $B^{- 1}$
(D) $(B^{- 1})^{'}$

Answer: (B)

Solution

From

B = A^{- 1} A^{'}

we get

B^{'} = (A^{- 1} A^{'})^{'} = A (A^{'})^{- 1}

. Then

B B^{'} = A^{- 1} A^{'} \cdot A (A^{'})^{- 1} = A^{- 1} (A^{'} A) (A^{'})^{- 1}

. The condition

A A^{'} = A^{'} A

lets us swap

A^{'} A

for

A A^{'}

, so

B B^{'} = A^{- 1} (A A^{'}) (A^{'})^{- 1} = (A^{- 1} A) (A^{'} (A^{'})^{- 1}) = I

. Hence

B B^{'} = I

.

JEE Main · 2014 · 06 Apr 2014 · Offline · Single correct option

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