For any $3 \times 3$ matrix $M$, let $|M|$ denote the determinant of $M$. Let $E = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{bmatrix}$, $P = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ and $F = \begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix}$ If $Q$ is a non-singular matrix of order $3 \times 3$, then which of the following statements is (are) TRUE?

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

$\text{Which of (1)--(4) are TRUE?}$

1. verify PEP = F and P²=I

$2.\ \text{find } |E| \text{ and } |F|$

$3.\ \text{test (2), (3), (4)}$

$P: \text{ swap rows/cols } 2 \leftrightarrow 3 \Rightarrow P^{2} = I,\quad P^{-1} = P$

PEP: swap R₂,R₃ then C₂,C₃ of E ⇒ F

$P^{2}=I \Rightarrow \text{Statement (1) TRUE}$

$R_3 = 2R_1 + 3R_2 \Rightarrow |E| = 0$

F = PEP ⇒ |F| = |P| |E| |P| = 0 PF = EP ⇒ |EQ + PFQ⁻¹| = |E(Q+PQ⁻¹)| = |E|·|Q+PQ⁻¹| = 0 |EQ| + |PFQ⁻¹| = 0 ⇒ Statement (2) TRUE

$|EF| = |E|\,|F| = 0 \Rightarrow |(EF)^3| = 0 = |EF|^2$

$0 > 0 \text{ is false} \Rightarrow \text{Statement (3) FALSE}$

$\operatorname{tr}(P^{-1}EP) = \operatorname{tr}(E),\quad \operatorname{tr}(P^{-1}FP) = \operatorname{tr}(F)$

$\text{both sides } = \operatorname{tr}(E) + \operatorname{tr}(F) \Rightarrow \text{Statement (4) TRUE}$

$\boxed{\text{Statements (1), (2) and (4) are TRUE}}$