# Use an inverse matrix to solve system of linear equations.x+2y=-1x-2y=3

Equations

Use an inverse matrix to solve system of linear equations.
$$x+2y=-1$$
$$x-2y=3$$

2021-02-19

Step 1
In matrix form, given system of equations cab be written as
$$x+2y = -1$$
$$x-2y=3$$
$$AX=B$$
$$\displaystyle{A}={\left[\begin{array}{cc} {1}&{2}\\{1}&-{2}\end{array}\right]},{X}={\left[\begin{array}{c} {x}\\{y}\end{array}\right]},{B}={\left[\begin{array}{c} -{1}\\{3}\end{array}\right]}$$
We know that
$$\displaystyle{\left[\begin{array}{cc} {a}&{b}\\{c}&{d}\end{array}\right]}^{{-{1}}}=\frac{{1}}{{{a}{d}-{b}{c}}}{\left[\begin{array}{cc} {d}&-{b}\\-{c}&{a}\end{array}\right]}$$
Step 2
Therefore,
$$\displaystyle{A}^{{-{1}}}=\frac{{1}}{{{1}{\left(-{2}\right)}-{2}{\left({1}\right)}}}{\left[\begin{array}{cc} -{2}&-{2}\\-{1}&{1}\end{array}\right]}$$
$$\displaystyle{A}^{{-{1}}}=-\frac{{1}}{{4}}{\left[\begin{array}{cc} -{2}&-{2}\\-{1}&{1}\end{array}\right]}$$
Therefore, solution of system of equations is
$$\displaystyle{X}={A}^{{-{1}}}{B}$$
$$\displaystyle{X}=-\frac{{1}}{{4}}{\left[\begin{array}{cc} -{2}&-{2}\\-{1}&{1}\end{array}\right]}{\left[\begin{array}{c} -{1}\\{3}\end{array}\right]}$$
$$\displaystyle{X}=-\frac{{1}}{{4}}{\left[\begin{array}{cc} {2}&-{6}\\{1}&{3}\end{array}\right]}{\left[\begin{array}{c} {1}\\-{1}\end{array}\right]}$$
$$x=1,y=-1$$
Result:$$x=1,y=-1$$