Can a 3 4 matrix have independent columns Independent rows

Can a 3 * 4 matrix have independent columns? Independent rows? Explain.

Solution

The rank of a matrix is defined as the number of linearly independent row vectors of the matrix. The rank of a matrix also equals the maximum number of linearly independent column vectors.Thus, the maximum possible rank of a 3x4 matrix is 3. An example of a 3 x 4 matrix with 3 independent rows is the matrix A with the rows (1, 0, 0,0), ( 0, 1, 0,0) and ( 0,0, 1, 0) . Its rank is 3. The three rows are linearly independent. Since the number of columns is 4, the columns of a 3 x 4 matrix cannot be linearly independent.