For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $E=\begin{aligned}…

For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $E=\begin{aligned} \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{bmatrix}, \end{aligned}$ $P=\begin{aligned} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}, \end{aligned}$ and $F=\begin{aligned} \begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix}. \end{aligned}$ If $Q$ is a nonsingular matrix of order $3 \times 3$, then which of the following statements is (are) TRUE?
  1. $F = PEP$ and $P^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$
  2. EQ+PFQ-1=|EQ|+PFQ-1
  3. (EF)3>|EF|2
  4. Sum of the diagonal entries of P-1EP+F is equal to the sum of diagonal entries of E+P-1FP

Solution

(1) $ P^2 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = I $ $ PEP = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} $ $ = \begin{bmatrix} 1 & 2 & 3 \\ 8 & 13 & 18 \\ 2 & 3 & 4 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix} $ $ = \begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix} = F $ (2) Here, $F = PEP \Rightarrow PF = P^2EP \Rightarrow PF = EP$, $|E| = 0$, $|F| = 0$ $ |EQ + PFQ^{-1}| = |EQ + EPQ^{-1}| = |E(Q + PQ^{-1})| = |E||Q + PQ^{-1}| = 0 $ $ |EQ| + |PFQ^{-1}| = 0 + 0 = 0 $ (3) Since, $|E| = 0 \Rightarrow |EF| = 0$

Thus, EF3>EF2 is not possible

(4) P=100001010⇒P=-1

P-1= Adj P|P|=--10000-10-10=100001010

P-1EP=10000101013224381813

=13281813243=F

F+P-1EP=F+F=2F=264163626486

Sum of Diagonal elements of F+P-1EP=44

E+P-1FP=E+10000101013281813243100001010

=12323481318+10000101012381318234

=12323481318+12323481318

=246468162636

TrE+P-1FP=44

Asked in: JEE Advanced 2021 (Paper 1)

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