Matrices a and B satisfy `AB=B^(-1)`, where `B=[(2,-1),(2,0)]`. Find
(i) without finding `B^(-1)`, the value of K for which
`KA-2B^(-1)+I=O`.
(ii) without finding `A^(-1)`, the matrix X satifying `A^(-1) XA=B`.

Explanation

(i) `AB=B^(-1)` or `AB B=B^(-1)B` or `AB^(2)=I`
Now, `KA-2B^(-1)+I=O`
or `KAB-2B^(-1)B+IB=O`
or `KAB-2I+B=O`
or `KAB^(2)-2B+B^(2)=O`
or `KI-2B+B^(2)=O`
or `K[(1,0),(0,1)]-2[(2,-1),(2,0)]+[(2,-1),(2,0)][(2,-1),(2,0)] =[(0,0),(0,0)]`
or `[(K,0),(0,K)]-[(4,-2),(4,0)]+[(2,-2),(4,-1)]=[(0,0),(0,0)]`
or `[(K-2,0),(0,K-2)]=[(0,0),(0,0)]`
ot `K=2`
(ii) `A^(-1)XA=B`
or `A A^(-1)XA=AB`
or `IXA=AB`
or `XAB=AB^(2)`
or `XAB=I`
or `XAB^(2)=B`
or `XI=B`
or `X=B`