Question

Prove that the inverse of `[[A,O],[B,C]]` is
`[[A^(-1),O],[-C^(-1)BA^(-1),C^(-1)]]` , where A, Care non-singular matrices and
O is null matrix and find the inverse. `[[1,0,0,0],[1,1,0,0],[1,1,1,0],[1,1,1,1]]`

Answer 1

We have, First part `[[A,O],[B,C]][[A^(-1),O],[-C^(-1)BA^(-1),C^(-1)]]`
`=[[A A^(-1),O],[BA^(-1)-C C ^(-1)BA^(-1),C C^(-1)]]`
`=[[I,O],[BA^(-1)-BA^(-1),I]]=[[I,O],[0,I]]`
Hence, `[[A^(-1),O],[-C^(-1)BA^(-1),C^(-1)]]` is the inverse of `[[A,O],[B,C]] `
Second part `[[1,0,0,0],[1,1,0,0],[1,1,1,0],[1,1,1,1]] = [[A,O],[B,C]]`
where `A = [[1,0],[1,1]], B=[[1,1],[1,1]],C=[[1,0],[1,1]]and O = [[0,0],[0,0]]`
and `A^(-) = [[1,0],[-1,1]],C^(-1)=[[1,0],[-1,1]]`
Now, `C^(-1)BA^(-1) = [[1,0],[-1,1]][[1,1],[1,1]][[1,0],[-1,1]] = [[0,0],[0,0]]`
`therefore` Inverse of `[[1,0,0,0],[1,1,0,0],[1,1,1,0],[1,1,1,1]] "is" [[1,0,0,0],[-1,1,0,0],[0,-1,1,0],[0,0,-1,1]]`