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Question

Let S be set of 2×2 matrices given by S={A=[abcd],where a, b, c, dI}such that AT=A1, then

A
Number of matrices in set S is equal to 8.
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B
Number of matrices in set S such that |AI2|0, is equal to 4.(where I2 is an identity matrix of order 2)
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C
Symmetric matrices are more than skew symmetric matrices in set S.
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D
All matrices in set S are singular.
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Solution

The correct options are
A Number of matrices in set S is equal to 8.
C Symmetric matrices are more than skew symmetric matrices in set S.
AT=A1
AAT=I
[abcd][acbd]=[1001]
a=0, b=±1, d=0, c=±1 ora=±1, b=0, d=±1, c=0
So total 8 matrices are possible.
[1001],[1001],[1001],[1001],[0110],[0110],[0110],[0110]
|AI2|=|AAAT|=|A||I2AT|=|A||(I2AT)T|=|A||I2A|
|AI2|=|A||I2A|
As |AI2|0|A|=1
So total 3 such matrices are possible
except 1 case
A=I=[1001] where |AI2|=0

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