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Question

If A=[1−123] and $B = \begin{bmatrix} 2& 3\\ -1 & -2\end{bmatrix}$, then which of the following is/are correct?
1.
AB(A−1B−1) is a unit matrix.
2. (AB)−1=A−1B−1
Select the correct answer using the code given below

A
1 only
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B
2 only
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C
Both 1 and 2
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D
Neither 1 nor 2
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Solution

The correct option is A 1 only
A=[1123]B=[2132]
detA=1detB=1
1.AB(A1B1)
ABA1B1
(AA1)(BB1)
=I×I
=I
For any value of A & B
AB(A1B1)=I for (detA&detB)0
1 is true
2.Consider AB(B1A1)
A(BB1)A1
AIA1
AA1=I
AB(B1A1)=I
B1A1=(AB)1
2is false
Option A

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