For any 3 × 3 matrix M, let $|M|$ denote the determinant of M. Let $E=\left[\begin{array}{ccc}1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18\end{array}\right], P=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right]$ and $F=\left[\begin{array}{ccc}1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3\end{array}\right]$ If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is (are) TRUE?

(A) $F=PEP$ and $P^{2}=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right]$; (B) $|EQ+PFQ^{-1}|=|EQ|+|PFQ^{-1}|$; (D) Sum of the diagonal entries of $P^{-1}EP+F$ is equal to the sum of diagonal entries of $E+P^{-1}FP$

Determinants — JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 Question with Solution

For any 3 × 3 matrix M, let

∣ M ∣

denote the determinant of M. Let

E = 12823133418, P = 100001010

and

F = 18231842133

If Q is a nonsingular matrix of order 3 × 3, then which of the following statements is (are) TRUE?

(A) $F = P E P$ and $P^{2} = 100010001$ ✓ Correct
(B) $∣ E Q + P F Q^{- 1} ∣ = ∣ E Q ∣ + ∣ P F Q^{- 1} ∣$ ✓ Correct
(C) $∣ (E F)^{3} ∣ > ∣ E F ∣^{2}$
(D)Sum of the diagonal entries of $P^{- 1} E P + F$ is equal to the sum of diagonal entries of $E + P^{- 1} F P$ ✓ Correct

Answer: (A), (B), (D)

Solution

(A) Direct computation shows

P E P = F

and

P^{2} = I

. (B) Since

∣ E ∣ = 0

and

∣ F ∣ = 0

, we have

∣ E Q ∣ = 0

and

∣ P F Q^{- 1} ∣ = 0

. For

T = E Q + P F Q^{- 1}

, we get

T Q = E Q^{2} + P F = E Q^{2} + P^{2} E P = E (Q^{2} + P)

, so

∣ T ∣∣ Q ∣ = ∣ E ∣∣ Q^{2} + P ∣ = 0

, thus

∣ T ∣ = 0

. The equation

0 = 0 + 0

holds. (C) Since

∣ E ∣ = 0

and

∣ F ∣ = 0

, we have

∣ E F ∣ = 0

, so

∣ (E F)^{3} ∣ = 0

and

∣ E F ∣^{2} = 0

. The inequality

0 > 0

is false. (D) Since

P^{- 1} = P

, we have

P^{- 1} F P = P F P = P^{2} E P = E

, so

E + P^{- 1} F P = 2 E

. Also,

P^{- 1} E P + F = P E P + F = 2 P E P

. The trace satisfies

Tr (2 P E P) = 2 Tr (E P P) = 2 Tr (E)

, which equals

Tr (2 E)

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · One or more correct options

JEE Advanced · 2021 · 03 Oct 2021 · Paper 1 · Determinants

Solution

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