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Question

# Let A be the set of all 3 × 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.The number of matrices A in A for which the system of linear equations A=⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣100⎤⎥⎦ has a unique solution is,

A
less than 4
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B
at least 4 but less than 6
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C
at least 7 but less than 10
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D
at least 10
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Solution

## The correct option is B at least 4 but less than 6⎡⎢⎣0aba0cbc1⎤⎥⎦either b=0 or c=0⇒|A|≠0⇒2 matrices⎡⎢⎣0aba1cbc0⎤⎥⎦either a=0 or c=0⇒|A|≠0⇒2 matrices⎡⎢⎣1abaocbco⎤⎥⎦either a=0 or b=0⇒|A|≠0⇒2 matrices.⎡⎢⎣1aba1cbc1⎤⎥⎦If a=b=0⇒|A|=0If a=c=0⇒|A|=0If b=c=0⇒|A|=0⇒ there will be only 6 matrices.

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