Solving a system of equations using the inverse of the coefficient matrix

Question

Let A be 2*2 matrix with A={{1 1}, {2 -2}}

use A^-1 to solve the matrix equation A{{x_1 x_2}} ={{3 5}}

I got A^-1 ={{1/2 1/4 },{1/2 -1/2}}

just need to know how to solve equation .

Answer 1

You can do this pretty easily by using Gaussian Elimination.

$$(A|b) =\begin{pmatrix} 1 & \ \ \ 1 & 3\\ 2 &-1 & 5 \end{pmatrix} \to^{(-2R_1+R_2)} \begin{pmatrix} 1 & \ \ \ 1 & \ \ \ 3\\ 0 & -3 & -1 \end{pmatrix} \Rightarrow x_1+x_2 = 3, -3x_2 = -2 $$

Hence, $x_2 = \frac{2}{3}, x_1= 3-\frac{2}{3}.$

Answer 2

Hint: What happens if you multiply both sides by $A^{-1}$? Your inverse is incorrect though.