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

$$4X + 5Y = 6$$

$$X + 5Y = 3$$

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

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

$R_1=R_1/4$

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

$R_2=R_2-R_1$

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

$R_2=4R_2/15$

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

$R_1=R_1-5R_2/4$

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

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