Question

How do you find the inverse of `A=``((0, 2, 4), (1, 3, 3), (1, 5, 8))`?

Answer 1

The inverse is `A^-1=((-9/2,-2,3),(5/2,2,-2),(-1,-1,1))`

Explanation:

Proceed as follows

Our matrix is `A=((0,2,4),(1,3,3),(1,5,8))`

Write side by side `A` and `I`

`((0,2,4),(1,3,3),(1,5,8))((1,0,0),(0,1,0),(0,0,1))`

Perform the following row operations

`R1harrR3`

`((1,5,8),(1,3,3),(0,2,4))((0,0,1),(0,1,0),(1,0,0))`

`R2larrR2-R1`

`((1,5,8),(0,-2,-5),(0,2,4))((0,0,1),(0,1,-1),(1,0,0))`

`R3larrR3+R2`

`((1,5,8),(0,-2,-5),(0,0,-1))((0,0,1),(0,1,-1),(1,1,-1))`

`R3larr(R3)xx(-1)`

`((1,5,8),(0,-2,-5),(0,0,1))((0,0,1),(0,1,-1),(-1,-1,1))`

`R2larr(R2)xx(-1)`

`((1,5,8),(0,2,5),(0,0,1))((0,0,1),(0,-1,1),(-1,-1,1))`

`R2larr(R2)-5(R3)`

`((1,5,8),(0,2,0),(0,0,1))((0,0,1),(5,4,-4),(-1,-1,1))`

`R1larr(R1)-8(R3)`

`((1,5,0),(0,2,0),(0,0,1))((8,8,-7),(5,4,-4),(-1,-1,1))`

`R2larr(R2)/2`

`((1,5,0),(0,1,0),(0,0,1))((8,8,-7),(5/2,2,-2),(-1,-1,1))`

`R1larr(R1)-5(R2)`

`((1,0,0),(0,1,0),(0,0,1))((-9/2,-2,3),(5/2,2,-2),(-1,-1,1))`

Therefore,

`A^-1=((-9/2,-2,3),(5/2,2,-2),(-1,-1,1))`