Elementary row and column transformations

 

Elementary row and column transformations are operations that can be performed on a matrix to transform it into an equivalent matrix, without changing its rank or its solutions. These transformations can be useful for simplifying matrices, solving systems of linear equations, and finding the inverse of a matrix.

There are three types of elementary row transformations:

1. Interchanging two rows.
2. Multiplying a row by a non-zero scalar.
3. Adding a scalar multiple of one row to another row.

There are also three types of elementary column transformations:

1. Interchanging two columns.
2. Multiplying a column by a non-zero scalar.
3. Adding a scalar multiple of one column to another column.

For example, consider the following matrix:

[a b c] [d e f] [g h i]

We can perform the following elementary row transformation:

1. Multiply the second row by 2:

[a b c] [2d 2e 2f] [g h i]

2. Add twice the second row to the third row:

[a b c] [2d 2e 2f] [g+2d h+2e i+2f]

We can also perform the following elementary column transformation:

1. Multiply the third column by 3:

[a b 3c] [d e 3f] [g h 3i]

2. Add the third column to the second column:

[a b+3c 3c] [d e+3f 3f] [g h+3i 3i]