Elementary row and column transformations

 

The rank of a matrix is a measure of the number of linearly independent rows or columns in the matrix. It is an important concept in linear algebra, and has many applications in engineering, science, and computer science.

There are several ways to calculate the rank of a matrix, but one common method is the row reduced echelon form (RREF). To put a matrix in RREF, you perform a series of elementary row operations to transform the matrix until it meets the following conditions:

1. The first non-zero element in each row, called the pivot, is 1.
2. The pivot is the only non-zero element in its column.
3. All the elements above and below the pivot are 0.

The rank of the matrix is equal to the number of pivots in the RREF.

For example, consider the following matrix:

[1 2 3] [0 4 5] [0 0 0]

To put this matrix in RREF, we can perform the following elementary row operations:

1. Divide the first row by 1 to make the pivot 1:

[1 2 3] [0 4 5] [0 0 0]

2. Subtract twice the first row from the second row to eliminate the element in the second row and first column:

[1 2 3] [0 0 1] [0 0 0]

The resulting matrix is in RREF, and has 2 pivots (1 and 0). Therefore, the rank of the matrix is 2