Based on how a system must work I noticed the following simplification of matrix inversion for a given pattern of zeros in rows and columns and I would like to know if there is general proof that says this is always true. Let me show an example of the simplification.
A = [ a1, a2, 0, 0, a5] [ a6, a7, 0, 0, a10] [ 0, 0, a13, a14, 0] [ 0, 0, a17, a18, 0] [ a21, a22, 0, 0, a25]
This matrix has zeros in third and fourth row and columns except for the elements within the cross of those zeros. This matrix can represent a system of which variables 3 and 4 are independent of the others. Therefore when one inverts B to solve such a system one must preserve that variables 3 and 4 are still independent. Therefore it follows that the zeros that exist in B will preserve in B^-1 in the same position. Moreover, one can perform the matrix inversion by splitting in two matrices such as:
A1 = [ a1, a2, a5] [ a6, a7, a10] [ a21, a22, a25] A2 = [ a13, a14] [ a17, a18]
And then just inverting them independently:
A1^-1 = [ b1, b2, b5] [ b6, b7, b10] [ b21, b22, b25] A2^-1 = [ b13, b14] [ b17, b18]
The original matrix inversion is formed by combining them:
A^-1 = [ b1, b2, 0, 0, b5] [ b6, b7, 0, 0, b10] [ 0, 0, b13, b14, 0] [ 0, 0, b17, b18, 0] [ b21, b22, 0, 0, b25]
My point being is that this property holds for any other combinations where the zeros make this happen (for example a 10x10 matrix with 4 other variables being independent of which 2 are together and 2 alone, etc.)
Is there a known proof somewhere for this property of matrix inversion?