If `B and C` are non-singular matrices and `O` is null matrix, then show that `[[A, B],[ C ,O]]^(-1)=[[O, C^(-1)],[B^(-1),-B^-1A C^(-1)]]dot`

Explanation

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]]`