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

Question

Which of the following is(are) NOT the square of a 3× 3 matrix with real entries? (A) [1&0&0 0&1&0 0&0&1] (B) [1&0&0 0&1&0 0&0&-1] (C) [1&0&0 0&-1&0 0&0&-1] (D) [-1&0&0 0&-1&0 0&0&-1]

Tardigrade Admin · Accepted Answer

A2=B | A |=± √|B| ∴ | B | ≠-1

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

$⎣ ⎡ 100010001 ⎦ ⎤$

$⎣ ⎡ 100010 00 - 1 ⎦ ⎤$

$⎣ ⎡ 100 0 - 1 0 00 - 1 ⎦ ⎤$

$⎣ ⎡ - 1 00 0 - 1 0 00 - 1 ⎦ ⎤$

$A^{2} = B$
$∣ A ∣ = \pm ∣ B ∣$
$∴ ∣ B ∣ \neq = - 1$

Q. Which of the following is(are) NOT the square of a 3×3 matrix with real entries?

⎣⎡​100​010​001​⎦⎤​

⎣⎡​100​010​00−1​⎦⎤​

⎣⎡​100​0−10​00−1​⎦⎤​

⎣⎡​−100​0−10​00−1​⎦⎤​