Rank of a matrix

Question 1

What is the rank of a matrix?

Answer 1

The rank is the maximum number of linearly independent rows or columns in a matrix. It represents the dimension of the column space (or row space), which corresponds to the number of unique solutions a system of linear equations has.

Question 2

How do you calculate the rank of a matrix?

Answer 2

1. Perform row-reduction to bring the matrix into row echelon form.
2. Count the number of non-zero rows.
   This count gives the rank of the matrix.

Question 3

Row Reduction:

Answer 3

Row reduction (also called Gaussian elimination) is a process used to simplify a matrix to a form that makes it easier to determine properties like its rank. The goal of row reduction is to transform the matrix into a simpler form called row echelon form (REF) or reduced row echelon form (RREF). This is done through a series of elementary row operations.

The three elementary row operations are:

1. Swap two rows: This swaps the positions of two rows in the matrix.
2. Multiply a row by a nonzero scalar: This scales a row by any non-zero value.
3. Add or subtract a multiple of one row from another row: This allows you to eliminate terms below (or above) the pivot positions.