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

*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,

*A* is a 3 x 5 matrix, this argument shows that

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,

The fact that the vectors **r** _{3} and **r** _{4} can be written as linear combinations of the other two ( **r** _{1} and **r** _{2}, 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 **a** _{11} = 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*— **c** _{1}, **c** _{2}, **c** _{3}, and **c** _{4}—are linearly independent. Show that this is indeed the case by verifying the relations

**c** _{1} and **c** _{3} 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:

Since there are 3 nonzero rows remaining in this echelon form of *B*,

**Example 2**: Determine the rank of the 4 by 4 checkerboard matrix

Since **r** _{2} = **r** _{4} = **−r** _{1} and **r** _{3} = **r** _{1}, all rows but the first vanish upon row‐reduction:

Since only 1 nonzero row remains, rank *C* = 1.