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JEE Advance - Mathematics (2009 - Paper 1 Offline - No. 20)

Let A be the set of all 3 $$\times$$ 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
Let A be the set of all 3 $$\times$$ 3 symmetric matrices all of whose entries are either 0 or 1. Five of these entries are 1 and four of them are 0.
Let A be the set of all 3 $$\times$$ 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 $$A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$$ is inconsistent, is
0
more than 2
2
1

Explanation

The six matrices A for which $$|A| = 0$$ are as follows:

$$\left[ {\matrix{ 0 & 0 & 1 \cr 0 & 0 & 1 \cr 1 & 1 & 1 \cr } } \right] \Rightarrow $$ inconsistent.

$$\left[ {\matrix{ 0 & 1 & 0 \cr 1 & 1 & 1 \cr 0 & 1 & 0 \cr } } \right] \Rightarrow $$ inconsistent.

$$\left[ {\matrix{ 1 & 1 & 1 \cr 1 & 0 & 0 \cr 1 & 0 & 0 \cr } } \right] \Rightarrow $$ infinite solutions.

$$\left[ {\matrix{ 1 & 1 & 0 \cr 1 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right] \Rightarrow $$ inconsistent.

$$\left[ {\matrix{ 1 & 0 & 1 \cr 0 & 1 & 0 \cr 1 & 0 & 1 \cr } } \right] \Rightarrow $$ inconsistent.

$$\left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 1 \cr 0 & 1 & 1 \cr } } \right] \Rightarrow $$ infinite solutions.

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