Mjollnir: Combined Row and Column Operations

Expanding the method

We saw on the previous page, A^{T^-1} = A^{-1^T}. Or A^{T^{-1^T}} = A^-1. And the inverse was obtained using column operations directly on the matrix without the need to take the transpose or do a transpose of the resulting inverse to get the solution to the problem. On this page we combine row and column operations, using the same square non-singular matrix.

Column and Row operations

We begin with the matrix from the previous page, augmented with an identity matrix of the appropriate rank for the operations and begin.

1	0	1
0	1	-1
1	0	-1

1	0	0
0	1	0
0	0	1

1	0	0
0	1	0
0	0	1

Subtract row 1 from row 3.

1	0	1
0	1	-1
0	0	-2

1	0	0
0	1	0
-1	0	1

1	0	0
0	1	0
0	0	1

Subtract column 1 from column 3.

1	0	0
0	1	-1
0	0	-2

1	0	0
0	1	0
-1	0	1

1	0	-1
0	1	0
0	0	1

Add column 2 to column 3.

1	0	0
0	1	0
0	0	-2

1	0	0
0	1	0
-1	0	1

1	0	-1
0	1	1
0	0	1

Divide through row 3 by -2.

1	0	0
0	1	0
0	0	1

1	0	0
0	1	0
1/2	0	-1/2

1	0	-1
0	1	1
0	0	1

Combine partial operations

To combine the column and row operations, we multiply the lower matrix with the right matrix. For convenience I'll refer to the lower matrix as C and the right matrix as R, with this layout:

I	R
C

Multiply the C and R:

1	0	-1
0	1	1
0	0	1

1	0	0
0	1	0
1/2	0	-1/2

1/2	0	1/2
1/2	1	-1/2
1/2	0	-1/2

The result is equivalent to that obtained in the previous section. But combining row and column operations leads to other possibilities we'll investigate in the next section.