Find a matrix B of order `2xx2` such that :
`[{:(1,1),(-2,3):}]B=[{:(3,4),(-1,2):}]`

Explanation

`LetA=[{:(1,1),(-2,3):}]" and "C=[{:(3,4),(-1,2):}]`
`therefore" "|A|=[{:(1,1),(-2,3):}]=3-(-2)=5ne0`
`therefore" A is invertible".`
Cofactors of martrix A
`c_(11)=(-1)^(1+1).3=3`
`c_(12)=(-1)^(1+2).(-2)=2`
`c_(21)=(-1)^(2+1).1=-1`
`c_(22)=(-1)^(2+2).1=1`
`therefore" adj.A"=[{:(c_(11),c_(12)),(c_(21),c_(22)):}]=[{:(3,2),(-1,1):}]=[{:(3,-1),(2,1):}]`
`"and "A^(-1)=1/(|A|)"adj.A"=1/5[{:(3,-1),(2,1):}]`
From eq. (1)
AB = C
`rArr" " A^(-1) (AB) = A^(-1) C`
`rArr" " (A^(-1)A)B=1/5[{:(3,-1),(2,1):}][{:(3,4),(-1,2):}]`
(From associative law)
`rArr" "IB=1/5[{:(9+1,12-2),(6-1,8+2):}](becauseA^(-1)A=I)`
`rArr" "B=[{:(2,2),(1,2):}]`