State the condition under which the following equations have a unique solution:

$x+2y-2z+5=0$

$-x+3y+4=0$

$-2y+z-4=0$.

Using Martin's rule, find the unique solution of the above system of linear equations.

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

$x+2y-3z=-4, 2x+3y+2z=2, 3x-3y-4z=11$.

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

$2x-3y+5z=11, 3x+2y-4z=-5, x+y-2z=-3$.

If $\mathrm{A}=\left[\begin{array}{cc}3& 5\\ -2& 3\end{array}\right]$, find ${\mathrm{A}}^{-1}$ and use it to solve the system of equations

$3x-2y=7, 5x+3y=1$.

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

$x+2y+z=4$,

$-x+y+z=0$,

$x-3y+z=2$.

If $\mathrm{A}=\left[\begin{array}{ccc}1& -2& 3\\ 1& 4& 1\\ 1& -3& 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}11& -5& -14\\ -1& -1& 2\\ -7& 1& 6\end{array}\right]$, find $\mathrm{AB}$ and use this result to solve the system of equations:

$x-2y+3z=6, x+4y+z=12, x-3y+2z=1$.

Using matrix method, solve the following system of homogeneous equations:

$5x+5y+2z=0$

$2x+5y+4z=0$

$4x+5y+2z=0$.

Using matrix method, solve the following system of homogeneous equations:

$3x+y-2z=0$

$x+y+z=0$

$x-2y+z=0$.