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

F= PEP and P 2 = [begin array lll 1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1end array ]

Sum of the diagonal entries of P -1 EP + F is equal to the sum of diagonal entries of E + P -1 FP

Mathematics · Matrices

JEE Advanced 2021 — Paper 1 — Question 23

For any $3 \times 3$ 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] \text { and } F=\left[\begin{array}{ccc}1 & 3 & 2 \\8 & 18 & 13 \\2 & 4 & 3\end{array}\right]

If Q is a non-singular matrix of order

$3 \times 3$ , then which of the following statements is(are) TRUE?

Option A:
$F=$ PEP and $P^{2}=\left[\begin{array}{lll}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{array}\right]$
Correct
Option B:
$\left|E Q+\mathrm{PFQ}^{-1}\right|=|E Q|+\left|\mathrm{PFQ}^{-1}\right|$
Correct
Option C:
$\left|(E F)^{3}\right|>|E F|^{2}$
Option D:
Sum of the diagonal entries of $\mathrm{P}^{-1} \mathrm{EP}+\mathrm{F}$ is equal to the sum of diagonal entries of $\mathrm{E}+\mathrm{P}^{-1} \mathrm{FP}$
Correct

Let

A = [C_{1} \; C_{2} \; C_{3}], \quad B = \begin{bmatrix} R_{1} \\ R_{2} \\ R_{3} \end{bmatrix}

AP = [C_{1} \; C_{3} \; C_{2}] \quad \text{and} \quad PB = \begin{bmatrix} R_{1} \\ R_{3} \\ R_{2} \end{bmatrix}

P^{2} = I

P(EP) = P \begin{bmatrix} 1 & 3 & 2 \\ 2 & 4 & 3 \\ 8 & 18 & 13 \end{bmatrix} = \begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix} = F

|E| = \begin{vmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{vmatrix} \overset{R_{3} \to R_{3} - 3R_{2} - 2R_{1}}{=} \begin{vmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 0 & 0 & 0 \end{vmatrix} = 0

\Rightarrow |F| = 0

|EQ + PFQ^{-1}| = |EQ + PPEPQ^{-1}| = |EQ + EPQ^{-1}| = |E| \, |Q + PQ^{-1}| = 0

|EQ| = |E| \, |Q| = 0, \quad |PFQ^{-1}| = |P| \, |F| \, |Q^{-1}| = 0

\text{(D)} \quad P^{-1}EP + F = PEP + F = 2F \quad (\text{as } P^{-1} = P)

E + P^{-1}FP = E + P^{-1}PEPP = 2E \quad (\text{trace}(E) = \text{trace}(F))

Answer key and solution verified before publishing.

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