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Question

# The number of 3× 3 matrices A whose entries are either 0 or 1 and for which the system A⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣100⎤⎥⎦ has exactly two distinct solutions,is

A
0
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B
291
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C
168
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D
2
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Solution

## The correct option is C 0Let A=⎡⎢⎣a1a2a3b1b2b3c1c2c3⎤⎥⎦where these elements are 0 or 1.Given A⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣100⎤⎥⎦⎡⎢⎣a1a2a3b1b2b3c1c2c3⎤⎥⎦⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣100⎤⎥⎦⎡⎢⎣a1x+a2y+a3zb1x+b2y+b3zc1x+c2y+c3z⎤⎥⎦=⎡⎢⎣100⎤⎥⎦⇒a1x+a2y+a3z=0b1x+b2y+b3z=0c1x+c2y+c3z=0⇒b1=b2=b3=c1=c2=c3=0a1x+a2y+a3z=0This cannot have exactly two distinct solutions , i.e. not possible .Hence, there is no such A matrix.

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