What are the application of matrices?

If $x=\alpha ,y=\beta$ and $z=\gamma$ is a solution of the matrix equation $LUX=B$, where

$L=\left[\begin{array}{ccc}1& 0& 0\\ 2& 1& 0\\ 0& -2& 1\end{array}\right],U=\left[\begin{array}{ccc}1& -2& 3\\ 0& -1& 6\\ 0& 0& 2\end{array}\right],X=\left[\begin{array}{l}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}3\\ 11\\ -8\end{array}\right]$

then the sum of $\alpha +\beta +\gamma$ is equal to

If $A=\left[\begin{array}{rrr}1& -1& 2\\ 0& 2& -3\\ 3& -2& 4\end{array}\right],B=\left[\begin{array}{rrr}-2& 0& 1\\ 9& 2& -3\\ 6& 1& -2\end{array}\right]$ and $B=A^{-1}$.

Using $A^{-1}$ solve the system linear equations given below.

$x-y+2z=1$

$2y-3z=1$

$3x-2y+4z=2$

If $\mathrm{A}=\left[\begin{array}{ccc}2& -3& 5\\ 3& 2& -4\\ 1& 1& -2\end{array}\right]$, find ${\mathrm{A}}^{-1}.$

By using ${\mathrm{A}}^{-1}$ solve the system of equations

$2x-3y+5z=11$

$3x+2y-4z=-5$

$x+y-2z=-3$.

Solve the following system of linear equations by matrix method:

$x-y+2z=7$

$3x+4y-5z=-5$

$2x-y+3z=12$.

Solve the system of linear equations by using matrix method.

$x-y+2z=7$

$3x+4y-5z=-5$

$2x-y+3z=12$

If $A$ is a scalar matrix with scalar $k\ne 0$, of order $3$ then $A^{-1}$ is 

Solve the following system of equations:

$\left[\begin{array}{lll}3& 0& 3\\ 2& 1& 0\\ 4& 0& 2\end{array}\right]\left[\begin{array}{l}x\\ y\\ z\end{array}\right]=\left[\begin{array}{l}8\\ 1\\ 4\end{array}\right]+\left[\begin{array}{c}2y\\ z\\ 3y\end{array}\right]$

Express the following equation in matrix form and solve them by the method of reduction

$x+y+z=6$, $3x-y+3z=10$ and $5x+5y-4z=3$

If $x=\alpha ,y=\beta$ and $z=\gamma$ is a solution of the system of linear equations

$x-2y+z=0$
$x-y+4z=11$
$x+2y+4z=17$

then the product $\alpha \beta \gamma$ is equal to

Find the adjoint of the matrix $A=\left[\begin{array}{rr}1& 2\\ 3& -5\end{array}\right]$ and verify the result $A\left(\mathrm{adj}A\right)=\left(\mathrm{adj}A\right)A=\left|A\right|·I$.

Solve the system of linear equations by matrix method

$\mathrm{x}-\mathrm{y}+\mathrm{z}=4$

$2\mathrm{x}+\mathrm{y}-3\mathrm{z}=0$

$x+y+z=2$

Solve the following equations by the method reduction:

$$\begin{gathered} x+3y+3z=12 \\ x+4y+4z=15 \\ x+3y+4z=13 \end{gathered}$$

Using matrices, solve the following system of equations:

$2x-3y+5z=11$

$3x+2y-4z=-5$

$x+y-2z=-3$