JEE MAIN - Mathematics (2010 - No. 10)
The number of $$3 \times 3$$ non-singular matrices, with four entries as $$1$$ and all other entries as $$0$$, is :
$$5$$
$$6$$
at least $$7$$
less than $$4$$
Explanation
$$\left[ {\matrix{
1 & {...} & {...} \cr
{...} & 1 & {...} \cr
{...} & {...} & 1 \cr
} } \right]\,\,$$ are $$6$$ non-singular matrices because $$6$$
blanks will be filled by $$5$$ zeros and $$1$$ one.
Similarly, $$\left[ {\matrix{ {...} & {...} & 1 \cr {...} & 1 & {...} \cr 1 & {...} & {...} \cr } } \right]\,\,$$ are $$6$$ non-singular matrices.
So, required cases are more than $$7,$$ non-singular $$3 \times 3$$ matrices.
blanks will be filled by $$5$$ zeros and $$1$$ one.
Similarly, $$\left[ {\matrix{ {...} & {...} & 1 \cr {...} & 1 & {...} \cr 1 & {...} & {...} \cr } } \right]\,\,$$ are $$6$$ non-singular matrices.
So, required cases are more than $$7,$$ non-singular $$3 \times 3$$ matrices.
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