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For a general square matrix m and arbitrary partition of it into conformable parts m={{a,b},{c,d}} (i.e., a and d are square matrices, and b and c have appropriate dimensions),

the formula for the inverse is

 m={{a,b},{c,d}}; e = d - c.Inverse[a].b; minv={{Inverse[a] + Inverse[a].b.Inverse[e].c.Inverse[a], -Inverse[a].b.Inverse[e]}, {-Inverse[e].c.Inverse[a], Inverse[e]}}; 

This can be used to define a function that takes a matrix and a partition specification and does the needed type and conformability checks before applying the formula.

The formula can be found in, for example, Review of Matrix Algebra.

For a general square matrix m and arbitrary partition of it into conformable parts m={{a,b},{c,d}} (i.e., a and d are square matrices, and b and c have appropriate dimensions),

the formula for the inverse (which can be found in, for example, Review of Matrix Algebra) is

 m={{a,b},{c,d}}; e = d - c.Inverse[a].b; minv={{Inverse[a] + Inverse[a].b.Inverse[e].c.Inverse[a], -Inverse[a].b.Inverse[e]}, {-Inverse[e].c.Inverse[a], Inverse[e]}}; 

This can be used to define a function that takes a matrix and a partition specification and does the needed type and conformability checks before applying the formula. You can use @Mike's formulation (and, off course, heed his warnings) as a first step:

Clear[sInverse2]; sInverse2[{{a_, b_}, {c_, d_}}] := With[{e = d - c.Inverse[a].b}, {{Inverse[a] + Inverse[a].b.Inverse[e].c.Inverse[a], -Inverse[a].b.Inverse[e]}, {-Inverse[e].c.Inverse[a], Inverse[e]}}] 

For a general square matrix m and arbitrary partition of it into conformable parts m={{a,b},{c,d}} (i.e., a and d are square matrices, and c and d have appropriate dimensions),

the formula for the inverse is

 m={{a,b},{c,d}}; e = d - c.Inverse[a].b; minv={{Inverse[a] + Inverse[a].b.Inverse[e].c.Inverse[a], -Inverse[a].b.Inverse[e]}, {-Inverse[e].c.Inverse[a], Inverse[e]}}; 

This can be used to define a function that takes a matrix and a partition specification and does the needed type and conformability checks before applying the formula.

The formula can be found in, for example, Review of Matrix Algebra.

For a general square matrix m and arbitrary partition of it into conformable parts m={{a,b},{c,d}} (i.e., a and d are square matrices, and b and c have appropriate dimensions),

the formula for the inverse is

 m={{a,b},{c,d}}; e = d - c.Inverse[a].b; minv={{Inverse[a] + Inverse[a].b.Inverse[e].c.Inverse[a], -Inverse[a].b.Inverse[e]}, {-Inverse[e].c.Inverse[a], Inverse[e]}}; 

This can be used to define a function that takes a matrix and a partition specification and does the needed type and conformability checks before applying the formula.

The formula can be found in, for example, Review of Matrix Algebra.

For a general square matrix m and arbitrary partition of it into conformable parts m={{a,b},{c,d}} (i.e., a and d are square matrices, and c and d have appropriate dimensions),

the formula for the inverse is

 m={{a,b},{c,d}}; e = d - c.Inverse[a].b; minv={{Inverse[a] + Inverse[a].b.Inverse[e].c.Inverse[a], -Inverse[a].b.Inverse[e]}, {-Inverse[e].c.Inverse[a], Inverse[e]}}; 

For a general square matrix m and arbitrary partition of it into conformable parts m={{a,b},{c,d}} (i.e., a and d are square matrices, and c and d have appropriate dimensions),

the formula for the inverse is

 m={{a,b},{c,d}}; e = d - c.Inverse[a].b; minv={{Inverse[a] + Inverse[a].b.Inverse[e].c.Inverse[a], -Inverse[a].b.Inverse[e]}, {-Inverse[e].c.Inverse[a], Inverse[e]}}; 

This can be used to define a function that takes a matrix and a partition specification and does the needed type and conformability checks before applying the formula.

The formula can be found in, for example, Review of Matrix Algebra.