Answer to Question #287796 in Linear Algebra for Zeki

Question

find the inverse of this matrix by using Gausian method

A= 4X + 5Y = 6

X + 5Y = 3

Answer 1

$4X + 5Y = 6$

$X + 5Y = 3$

$A=\begin{bmatrix} 4 & 5 \\ 1 & 5 \end{bmatrix}$

$\begin{bmatrix} 4 & 5 & & 1 & 0 \\ 1 & 5 & & 0 & 1 \end{bmatrix}$

$R_1=R_1/4$

$\begin{bmatrix} 1 & 5/4 & & 1/4 & 0 \\ 1 & 5 & & 0 & 1 \end{bmatrix}$

$R_2=R_2-R_1$

$\begin{bmatrix} 1 & 5/4 & & 1/4 & 0 \\ 0 & 15/4 & & -1/4 & 1 \end{bmatrix}$

$R_2=4R_2/15$

$\begin{bmatrix} 1 & 5/4 & & 1/4 & 0 \\ 0 & 1 & & -1/15 & 4/15 \end{bmatrix}$

$R_1=R_1-5R_2/4$

$\begin{bmatrix} 1 & 0 & & 1/3 & -1/3 \\ 0 & 1 & & -1/15 & 4/15 \end{bmatrix}$

$A^{-1}=\begin{bmatrix} 1/3 & -1/3 \\ -1/15 & 4/15 \end{bmatrix}$