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JEE Advance - Mathematics (2009 - Paper 1 Offline - No. 19)

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 $$A\left[ {\matrix{ x \cr y \cr z \cr } } \right] = \left[ {\matrix{ 1 \cr 0 \cr 0 \cr } } \right]$$ has a unique solution, is
less than 4
at least 4 but less than 7
at least 7 but less than 10
at least 10

Explanation

We have

$$\left[ {\matrix{ 0 & a & b \cr a & 0 & c \cr b & c & 1 \cr } } \right]$$

We can see that either $$b = 0$$ or $$c = 0 \Rightarrow |A| \ne 0$$; therefore two matrices.

$$\left[ {\matrix{ 0 & a & b \cr a & 1 & c \cr b & c & 0 \cr } } \right]$$

Now, either $$a = 0$$ or $$c = 0 \Rightarrow |A| \ne 0$$; therefore, two matrices

$$\left[ {\matrix{ 1 & a & b \cr a & 0 & c \cr b & c & 0 \cr } } \right]$$

Now, either $$a = 0$$ or $$b = 0 \Rightarrow |A| \ne 0$$; therefore two matrices.

$$\left[ {\matrix{ 1 & a & b \cr a & 1 & c \cr b & c & 1 \cr } } \right]$$

When $$a = b = 0 \Rightarrow |A| = 0$$

When $$a = c = 0 \Rightarrow |A| = 0$$

When $$b = c = 0 \Rightarrow |A| = 0$$

Therefore, there are only six matrices.

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