JEE Advance - Mathematics (2017 - Paper 1 Offline - No. 6)

Which of the following is(are) NOT the square of a 3 $$ \times $$ 3 matrix with real entries?

$$\left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & { - 1} \cr } } \right]$$

$$\left[ {\matrix{ 1 & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right]$$

$$\left[ {\matrix{ { - 1} & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right]$$

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

For a matrix to be square of matrix with real entries, its determinant should be positive.

Option (A) : $$\left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & { - 1} \cr } } \right]$$, determinant is NOT possible:

1($$-$$1) $$-$$ 0(0) + 0(0) = $$-$$1

Option (B) : $$\left[ {\matrix{ 1 & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right]$$, determinant is possible:

1(1) $$-$$ 0(0) + 0(0) = +1

Option (C) : $$\left[ {\matrix{ { - 1} & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right]$$, determinant is NOT possible :

$$-$$1(1) $$-$$ 0(0) + 0(0) = $$-$$1

Option (D) : $$\left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right]$$, determinant is possible:

(1) $$-$$ 0(0) + 0(0) = +1

Thus, options (A) and (C) are NOT the square of a 3 $$\times$$ 3 matrix with real entries.