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. 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. 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 second equation above says that similar operations performed on the fourth row can produce a row of zeros there also. 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 equations in (***) can be rewritten as follows: Thus, the row rank-and therefore the rank-of this matrix is 2. 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. The process by which the rank of a matrix is determined can be illustrated by the following example. So, if A is a 3 x 5 matrix, this argument shows that

Thus, the column rank-and therefore the rank-of such a matrix can be no greater than 3. Any collection of more than three 3‐vectors is automatically dependent. Although three 5‐vectors could be linearly independent, it is not possible to have five 3‐vectors that are independent. A 3 x 5 matrix,Ĭan be thought of as composed of three 5‐vectors (the rows) or five 3‐vectors (the columns). 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. Where min( m, n) denotes the smaller of the two numbers m and n (or their common value if m = n). Therefore, if A is m x n, it follows from the inequalities in (*) that What is not so obvious, however, is that for any matrix A,īecause 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.

Matrix a^2