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JEE Advanced 2017 · Determinants

Which of the following is(are) not the square of a 3×33 \times 3 matrix with real entries?

Answers: B. [10001000−1]\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix} · D. [−1000−1000−1]\begin{bmatrix} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}

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Written solution

Options are diagonal: diag⁡(1,1,1), diag⁡(1,1,−1), diag⁡(1,−1,−1), diag⁡(−1,−1,−1)\text{Options are diagonal: } \operatorname{diag}(1,1,1),\ \operatorname{diag}(1,1,-1),\ \operatorname{diag}(1,-1,-1),\ \operatorname{diag}(-1,-1,-1)

To find: which A is not B2 for any real 3×3 matrix B\text{To find: which } A \text{ is not } B^2 \text{ for any real } 3\times 3 \text{ matrix } B

A=B2  ⇒  det⁡A=(det⁡B)2≥0A=B^2 \;\Rightarrow\; \det A=(\det B)^2\ge 0

Plan: use det⁡; then verify the rest really are squares\text{Plan: use } \det; \text{ then verify the rest really are squares}

det⁡(diag⁡(d1,d2,d3))=d1d2d3\det(\operatorname{diag}(d_1,d_2,d_3))=d_1d_2d_3

(1) det⁡=1;(2) det⁡=−1(1)\ \det=1;\quad (2)\ \det=-1

(3) det⁡=1;(4) det⁡=−1(3)\ \det=1;\quad (4)\ \det=-1

det⁡=−1<0  ⇒  (2), (4) cannot be squares\det=-1<0 \;\Rightarrow\; (2),\ (4) \text{ cannot be squares}

I=I2  ⇒  (1) is a squareI=I^2 \;\Rightarrow\; (1)\ \text{is a square}

(3)=1 0 0 R, R²=-I₂ R=0 -1 1 0 R²=-1 0 0 -1=-I₂ B=1 0 0 0 0 -1 0 1 0 ⇒ B²=diag(1,-1,-1)

(2) det⁡=−1<0  ⇒  not a square(2)\ \det=-1<0 \;\Rightarrow\; \text{not a square}

(4) −I3, det⁡=−1<0  ⇒  not a square(4)\ -I_3,\ \det=-1<0 \;\Rightarrow\; \text{not a square}

Not squares: (2) and (4)\text{Not squares: } (2)\ \text{and}\ (4)

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