Question

Solve the following systems of simultaneous equations using the inverse matrix of Gauss Jordan X1+2x2+3x3=3? 2x1+4x2+5x3=4? 3x1+5x2+6x3=8?

Answer 1

The solution is `((x_1),(x_2),(x_3))=((7),(-5),(2))`

Explanation:

The augmented matrix is

`A=((1,2,3,|,3),(2,4,5,|,4),(3,5,6,|,8))`

The main matrix is

`A_1=((1,2,3),(2,4,5),(3,5,6))`

The inverse is calculated as follows

Write side by side `A` and `I_3` on the right

`((1,2,3),(2,4,5),(3,5,6))((1,0,0),(0,1,0),(0,0,1))`

Perform the row operations

`R2larrR2-2xxR1` and `R3larrR3-3xxR1`

`((1,2,3),(0,0,-1),(0,-1,-3))((1,0,0),(-2,1,0),(-3,0,1))`

`R3harrR2`

`((1,2,3),(0,-1,-3),(0,0,-1))((1,0,0),(-3,0,1),(-2,1,0))`

`R2larr(R2)/(-1)` and `R3larr(R3)/(-1)`

`((1,2,3),(0,1,3),(0,0,1))((1,0,0),(3,0,-1),(2,-1,0))`

`R1larrR1-3xxR3` and `R2larrR2-3xxR3`

`((1,2,0),(0,1,0),(0,0,1))((-5,3,0),(-3,3,-1),(2,-1,0))`

`R1larrR1-2xxR2`

`((1,0,0),(0,1,0),(0,0,1))((1,-3,2),(-3,3,-1),(2,-1,0))`

Therefore,

`A_1^-1=((1,-3,2),(-3,3,-1),(2,-1,0))`

Then,

`((x_1),(x_2),(x_3))=((1,-3,2),(-3,3,-1),(2,-1,0))*((3),(4),(8)`

`=((7),(-5),(2))`