Solve the following equations by the method of inversion :
` x + 2y = 2, 2x + 3y = 3`.

Explanation

The given equations can be written in the matrix form as :
`{:((1,2),(2,3)):} {:((x),(y)):}= {:((2),(3)):}`
This is of the form AX = B, where
Let us fiind `A^(-1)`
`|A| = |(1,2),(2,3)| = 3 - 4 = - 1 ne 0`
` :. A^(-1)` exists.
Consider `A A^(-1) = I`
`:. {:((1,2),(2,3)):} A^(-1) = {:((1,0),(0,1)):}`
By `R_(2) - 2R_(1) `, we get,
`{:((1,2),(0,-1)):} A^(-1) = {:((1,0),(-2,1)):}`
By `(-1)R_(2)` we get ,
`{:((1,2),(0,1)):}A^(-1)={:((1,0),(2,-1)):}`
By `R_(1) - 2R_(2)`, we get,
`{:((1,0),(0,1)):} A^(-1) = {:((-3,2),(2,-1)):}`
` :. A^(-1) = {:((-3,2),(2,-1)):}`
Now, premultiply `AX = B " by " A^(-1) `, we get ,
` A^(-1) (AX) = A^(-1) B`
` :. (A^(-1)A) X = A^(-1) B`
` :. IX = A^(-1) B`
`:. X = {:((-3,2),(2,-1)):}{:((2),(3)):}`
`:. {:((x),(y)):}={:((-6+6),(4-3)):}={:((0),(1)):}`
By equality of matrices,
x = 0, y = 1 is the required solution.