Programming + Errors: The Rank of a Matrix

The maximum number of linearly independent rows in a matrix A is called the row rank of A, and the maximum number of linarly independent columns in A is called the column rank of A. If A is an m by n matrix, that is, if A has m rows and n columns, then it is obvious that

What is not so obvious, however, is that for any matrix A,

the row rank of A = the column rank of A

Because of this fact, there is no reason to distinguish between row rank and column rank; the common value is simply called the rank of the matrix. Therefore, if A is m x n, it follows from the inequalities in (*) that

where min( m, n) denotes the smaller of the two numbers m and n (or their common value if m = n). For example, the rank of a 3 x 5 matrix can be no more than 3, and the rank of a 4 x 2 matrix can be no more than 2. A 3 x 5 matrix,

can be thought of as composed of three 5-vectors (the rows) or five 3-vectors (the columns). Although three 5-vectors could be linearly independent, it is not possible to have five 3-vectors that are independent. Any collection of more than three 3-vectors is automatically dependent. Thus, the column rank—and therefore the rank—of such a matrix can be no greater than 3. So, if A is a 3 x 5 matrix, this argument shows that

in accord with (**). The process by which the rank of a matrix is determined can be illustrated by the following example. Suppose A is the 4 x 4 matrix

The four row vectors,

are not independent, since, for example

The fact that the vectors r₃ and r₄ can be written as linear combinations of the other two ( r₁ and r₂, which are independent) means that the maximum number of independent rows is 2. Thus, the row rank—and therefore the rank—of this matrix is 2.
The equations in (***) can be rewritten as follows:

The first equation here implies that if −2 times that first row is added to the third and then the second row is added to the (new) third row, the third row will be become 0, a row of zeros. The second equation above says that similar operations performed on the fourth row can produce a row of zeros there also. If after these operations are completed, −3 times the first row is then added to the second row (to clear out all entires below the entry a₁₁ = 1 in the first column), these elementary row operations reduce the original matrix A to the echelon form

The fact that there are exactly 2 nonzero rows in the reduced form of the matrix indicates that the maximum number of linearly independent rows is 2; hence, rank A = 2, in agreement with the conclusion above. In general, then, to compute the rank of a matrix, perform elementary row operations until the matrix is left in echelon form; the number of nonzero rows remaining in the reduced matrix is the rank. [Note: Since column rank = row rank, only two of the four columns in A— c₁, c₂, c₃, and c₄—are linearly independent. Show that this is indeed the case by verifying the relations

(and checking that c₁ and c₃ are independent). The reduced form of A makes these relations especially easy to see.] Example 1: Find the rank of the matrix

First, because the matrix is 4 x 3, its rank can be no greater than 3. Therefore, at least one of the four rows will become a row of zeros. Perform the following row operations: