For any 3 × 3 matrix M , let | M | denote the determinant of M ....

(1)

{P}^{2} = \left[\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}\right]\left[\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}\right] = \left[\begin{cases}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{cases}\right] = I

PEP = \left[\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}\right]\left[\begin{cases}1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18\end{cases}\right]\left[\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}\right]

= \left[\begin{cases}1 & 2 & 3 \\ 8 & 13 & 18 \\ 2 & 3 & 4\end{cases}\right]\left[\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}\right]

= \left[\begin{cases}1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3\end{cases}\right] = F

(2)

Here, $F = PEP ⇒ PF = {P}^{2}EP ⇒ PF = EP$, \left|E\right| = 0, \left|F\right| = 0$</p><p>\left|EQ + PF{Q}^{- 1}\right| = \left|EQ + EP{Q}^{- 1}\right| = ∣ E\left(Q + P{Q}^{- 1} | = | E | | Q + P{Q}^{- 1} ∣ = 0\right)</p><p>$| EQ | + \left|PF{Q}^{- 1}\right| = 0 + 0 = 0$</p><p>(3)</p><p>Since,$| E | = 0 ⇒ | EF | = 0$</p><p>Thus, \left|{\left(EF\right)}^{3}\right| > {\left|EF\right|}^{2} is not possible</p><p>(4)$P = \left[$\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}$\right] ⇒ \left|P\right| = - 1$</p><p>${P}^{- 1} = \dfrac{AdjP}{| P |} = - \left[$\begin{cases}- 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & - 1 & 0\end{cases}$\right] = \left[$\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}$\right]$</p><p>${P}^{- 1}EP = \left[$\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}$\right]\left[$\begin{cases}1 & 3 & 2 \\ 2 & 4 & 3 \\ 8 & 18 & 13\end{cases}$\right]$</p><p>$= \left[$\begin{cases}1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3\end{cases}$\right] = F$</p><p>$F + {P}^{- 1}EP = F + F = 2F = \left[$\begin{cases}2 & 6 & 4 \\ 16 & 36 & 26 \\ 4 & 8 & 6\end{cases}$\right]$</p><p>Sum of Diagonal elements of$F + {P}^{- 1}EP = 44$</p><p>$E + {P}^{- 1}FP = E + \left[$\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}$\right]\left[$\begin{cases}1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3\end{cases}$\right]\left[$\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}$\right]$</p><p>$= \left[$\begin{cases}1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18\end{cases}$\right] + \left[$\begin{cases}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{cases}$\right]\left[$\begin{cases}1 & 2 & 3 \\ 8 & 13 & 18 \\ 2 & 3 & 4\end{cases}$\right]$</p><p>$= \left[$\begin{cases}1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18\end{cases}$\right] + \left[$\begin{cases}1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18\end{cases}$\right]$</p><p>$= \left[$\begin{cases}2 & 4 & 6 \\ 4 & 6 & 8 \\ 16 & 26 & 36\end{cases}$\right]$</p><p>$Tr \left(E + {P}^{- 1}FP\right) = 44