Pensieve header: Whirling matrices until they get inverted.

CF=ExpandNumerator@*ExpandDenominator@*Together

MatrixForm[A=Table[(Subscript[a, 10i+j]),{i,3},{j,3}]]

MatrixForm[Ξ=A[[1;;-2,1;;-2]]]

θ=A[[-1,1;;-2]]

ϕ=A[[1;;-2,-1]]

α=A[[-1,-1]]

Outer[Times,ϕ,θ]//MatrixForm

W[A_?MatrixQ] := Module[{
Ξ=A[[1;;-2,1;;-2]],
θ=A[[-1,1;;-2]],
ϕ=A[[1;;-2,-1]],
α=A[[-1,-1]]
},
CF@ArrayFlatten[(({{{{(α^(-1))}}, {-(α^(-1))θ}}, {List /@ ((α^(-1))ϕ), Ξ-(α^(-1))Outer[Times,ϕ,θ]}}))]
]

W[A]//MatrixForm

A//W//W//MatrixForm

A//W//W//W//MatrixForm

Inverse[A]//MatrixForm

CF[(A//W//W//W)-Inverse[A]]

n=5;
MatrixForm[B=Table[(Subscript[a, 10i+j]),{i,n},{j,n}]]

CF[Nest[W,B,n]-Inverse[B]]

n=10;
MatrixForm[B=Table[RandomInteger[{-1,1}10^5],{i,n},{j,n}]]

W[B]

CF[Nest[W,B,n]-Inverse[B]]

Inverse[B]//N//MatrixForm

N@Log[10,Sqrt@Tr[Numerator[#].Transpose@Numerator[#]]& /@ NestList[W,B,n]]

N@Log[10,Sqrt@Tr[Denominator[#].Transpose@Denominator[#]]& /@ NestList[W,B,n]]

Y[A_?MatrixQ] := Module[{
Ξ=A[[1;;-2,1;;-2]],
θ=A[[-1,1;;-2]],
ϕ=A[[1;;-2,-1]],
α=A[[-1,-1]]
},
CF@ArrayFlatten[(({{{{1}}, {-θ}}, {List /@ (ϕ), α Ξ-Outer[Times,ϕ,θ]}}))]
]

n=4;
MatrixForm[B=Table[RandomInteger[{-1,1}n],{i,n},{j,n}]]

Nest[Y,B,n].B