JEE Advanced 2017 · Paper 1 · Mathematics Q5

MathematicsMultiple correct+4 / -2Question 41 of 54

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

\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}

\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}

\begin{bmatrix} 1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}

\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}

Answer and worked solution

Official answer: B, D

Key fact. $\det(M^2)=(\det M)^2\ge0$ for real $M$ .

(A) $I=I^2$ . It is a square.
(B) The determinant is $-1<0$ . Not a square.
(C) This one is a square: with $R=\begin{pmatrix}1&0&0\\0&0&-1\\0&1&0\end{pmatrix}$ (a quarter turn in the $y$ - $z$ plane), $R^2=\operatorname{diag}(1,-1,-1)$ .
(D) The determinant is $-1<0$ . Not a square.

(B), (D)

Topics:Determinants and properties Matrix algebra

JEE Advanced 2017 · Paper 1, Mathematics question 5