1
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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⎤⎥⎦ is inconsistent is

A
0
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B
more than 2
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C
2
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D
1
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Solution

## The correct option is B more than 2Let x be the number of 1's on the main diagonal and y be the number of 1's above the main diagonal, then x+2y=5⇒x=1,y=2orx=3,y=1.When x=1, the main diagonal can be chosen in 3 ways, and the elements above the main diagonal in 3 ways. Therefore, there are 9 such matrices.These are A1=⎡⎢⎣111100100⎤⎥⎦, A2=⎡⎢⎣011110100⎤⎥⎦ A3=⎡⎢⎣011100101⎤⎥⎦, A4=⎡⎢⎣101001110⎤⎥⎦ A5=⎡⎢⎣001011110⎤⎥⎦, A6=⎡⎢⎣001001111⎤⎥⎦A7=⎡⎢⎣110101010⎤⎥⎦, A8=⎡⎢⎣010111010⎤⎥⎦A9=⎡⎢⎣100011011⎤⎥⎦ When x=3, the main diagonal can be chosen in 3 way; and the element above the main diagonal in 3 ways. Therefore, there are 3 such matrices.A10=⎡⎢⎣110110001⎤⎥⎦, A11=⎡⎢⎣101010101⎤⎥⎦,A12=⎡⎢⎣100011011⎤⎥⎦Let X=⎡⎢⎣xyz⎤⎥⎦ and B=⎡⎢⎣100⎤⎥⎦, and note thatA1X=B and A12X=B have infinite number of solutions.AX=B has no solution whenA=A6,A8,A10,A11

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