How to invert a particular submatrix?

Suppose I have a matrix of numbers like this one in cells F2 to I5:

a_1 a_2 a_3 a_4 xb_1 b_2 b_3 b_4 yc_1 c_2 c_3 c_4 zd_1 d_2 d_3 d_4 w

Suppose that I want to solve the system of equations given by

a_1 a_3 a_4 xc_1 c_3 c_4 zd_1 d_3 d_4 w

Notice that the second matrix is the first one once we eliminate the second row and the second column.

I want to solve the system of equations given by the second matrix using MMULT and MINVERSE. However, MINVERSE does not let me select the values that I need from the first matrix, for it requires an array as input.

One option is to manually write the second matrix out of the first one and solve accordingly. While this works, it seems cumbersome. Is there anyway to use MMULT and MINVERSE to solve the system of equations given by the second matrix without having to write it explicitly (that is, telling MINVERSE which cells to pick from the first matrix).

I have also tried the sorting feature, but that breaks down many of the formulas in my spreadsheet, so it’s not an option.

So I guess that my question is: how can I invert a particular submatrix?