(a) If A = [2, -3, 5; 3, 2, -4; 1, 1, -2], find A^-1 and hence solve the following system of linear equations: 2 x-3 y +5 z =11, 3 x+2 y -4 z =-5, x+…

If $\displaystyle \mathrm{A}=\left[\begin{array}{ccc}2 & -3 & 5 \\ 3 & 2 & -4 \\ 1 & 1 & -2\end{array}\right]$, find $\displaystyle \mathrm{A}^{-1}$ and hence solve the following system of linear equations :

$\displaystyle 2 x-3 \mathrm{y}+5 \mathrm{z}=11, 3 x+2 \mathrm{y}-4 \mathrm{z}=-5, x+\mathrm{y}-2 \mathrm{z}=-3$

Using properties of determinants, prove that

$$\Delta=\left|\begin{array}{ccc}(\mathrm{b}+\mathrm{c})^2 & \mathrm{a}^2 & \mathrm{a}^2 \\ \mathrm{b}^2 & (\mathrm{c}+\mathrm{a})^2 & \mathrm{b}^2 \\ \mathrm{c}^2 & \mathrm{c}^2 & (\mathrm{a}+\mathrm{b})^2\end{array}\right|=2 \mathrm{abc}(\mathrm{a}+\mathrm{b}+\mathrm{c})^3$$