(a) If P = [1, -1, 0; 2, 3, 4; 0, 1, 2] and Q = [2, 2, -4; -4, 2, -4; 2, -1, 5], find (QP) and hence solve the following system of equations using…

If $\displaystyle \mathrm{P}=\left[\begin{array}{ccc}1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2\end{array}\right]$ and $\displaystyle \mathrm{Q}=\left[\begin{array}{ccc}2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5\end{array}\right]$, find (QP) and hence solve the following system of equations using matrices : \[x-\mathrm{y}=3,2 x+3 \mathrm{y}+4 \mathrm{z}=17, \mathrm{y}+2 \mathrm{z}=7 \]

Obtain the value of $\displaystyle \Delta=\left|\begin{array}{ccc}1+x & 1 & 1 \\ 1 & 1+\mathrm{y} & 1 \\ 1 & 1 & 1+\mathrm{z}\end{array}\right|$ in terms of $\displaystyle x, \mathrm{y}$ and z . Further, if $\displaystyle \Delta=0$ and $\displaystyle x, \mathrm{y}, \mathrm{z}$ are non-zero real numbers, prove that $\displaystyle x^{-1}+\mathrm{y}^{-1}+\mathrm{z}^{-1}=-1$.