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Question

If A is a non-singular matrix, then

A
A1 is symmetric if A is symmetric
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B
A1 is skew-symmetric if A is symmetric
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C
A1=|A|
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D
A1=|A|1
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Solution

The correct options are
A A1 is symmetric if A is symmetric
D A1=|A|1
Since |A|0, therefore A1 exists.
Now, AA1=I=A1A
(AA1)=I=(A1A)(A1)A=I=A(A1)(A=A)
(A1)A=I=A(A1)A1=(A1)A1 is symmetric
Also, since |A|0,A1 exists such that
AA1=I=A1AAA1=|I|
|A|A1=1(|AB|=|A||B|)
A1=1|A|

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