8.3. THE RANK OF A MATRIX 155

8.3 The Rank Of A Matrix8.3.1 The Definition Of RankTo begin, here is a definition to introduce some terminology.

Definition 8.3.1 Let A be an m× n matrix. The column space of A is the span of thecolumns. The row space is the span of the rows.

There are three definitions of the rank of a matrix which are useful. These are givenin the following definition. It turns out that the concept of determinant rank is oftenimportant but is virtually impossible to find directly. The other two concepts of rank arevery easily determined and it is a happy fact that all three yield the same number. This isshown later.

Definition 8.3.2 A sub-matrix of a matrix A is a rectangular array of numbers obtained bydeleting some rows and columns of A. Let A be an m×n matrix. The determinant rank ofthe matrix equals r where r is the largest number such that some r× r sub-matrix of A hasa non zero determinant. The row space of a matrix is the span of the rows and the columnspace of a matrix is the span of the columns. The row rank of a matrix is the number ofnonzero rows in the row reduced echelon form and the column rank is the number columnsin the row reduced echelon form which are one of the ek vectors. Thus the column rankequals the number of pivot columns. It follows the row rank equals the column rank. Thisis also called the rank of the matrix. The rank of a matrix A is denoted by rank(A) .

Example 8.3.3 Consider the matrix (1 2 32 4 6

)

What is its rank?

You could look at all the 2×2 submatrices(1 22 4

),

(1 32 6

),

(2 34 6

).

Each has determinant equal to 0. Therefore, the rank is less than 2. Now look at the 1×1submatrices. There exists one of these which has nonzero determinant. For example (1)has determinant equal to 1 and so the rank of this matrix equals 1.

Of course this example was pretty easy but what if you had a 4×7 matrix? You wouldhave to consider all the 4× 4 submatrices and then all the 3× 3 submatrices and then allthe 2× 2 matrices and finally all the 1× 1 matrices in order to compute the rank. Clearlythis is not practical. The following theorem will remove the difficulties just indicated.

The following theorem is proved later.

Theorem 8.3.4 Let A be an m×n matrix. Then the row rank, column rank and determinantrank are all the same.