Question

How to find the final form of augmented matrix when solving for the inverse of this system of equations? `2x+3y=4` `4x+5y=5`

A) `[(1,0),(0,1)]|[(5/2,-3/2),(-3,4)]`

B) `[(1,0),(0,1)]|[(-5/2,3/2),(2,-1)]`

C) `[(1,0),(0,1)]|[(5,-3),(-4,2)]`

D) `[(1,0),(0,1)]|[(5/2,-3/2),(-2,1)]`

I have zero clue of what the question wants me to do and I've never heard of finding the inverse of an augmented matrix.... Plz halp :/

Answer 1

The answer is `option (B)`

Explanation:

You have `2` equations with `2` unknowns

`{(2x+3y=4),(4y+5y=5):}`

The equations can be written as

`AX=B`

where `A` is a matrix

`((2,3),(4,5))((x),(y))=((4),(5))`

The solution is

`X=A^-1B`

Where `A^-1` is the inverse of matrix `A`

To calculate the inverse, proceed as follows,

Write side by side matrix `A` and the unit matrix `I`

Perform row operations until the unit matrix `I` is on the left and the inverse on the RHS

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

`R2larr( R2-2R1)`

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

`R2larr(R2*-1)`

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

`R1larrR1-3R2`

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

`R1larr((R1)/2)`

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

The answer is `option (B)`