The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.Consider matrix A and its row echelon matrix, Aref. Previously, we showed how to find the row echelon form for matrix A.
Because the row echelon form Aref has two non-zero rows, we know that matrix A has two independent row vectors; and we know that the rank of matrix A is 2.You can verify that this is correct. Row 1 and Row 2 of matrix A are linearly independent. However, Row 3 is a linear combination of Rows 1 and 2. Specifically, Row 3 = 3*( Row 1 ) + 2*( Row 2). Therefore, matrix A has only two independent row vectors.