Question

If matrix A has rank r, find nullity A, rank A, and rank A transpose.

a) Nullity A = r, Rank A = r, Rank A^T = r
b) Nullity A = 0, Rank A = r, Rank A^T = r
c) Nullity A = 0, Rank A = 0, Rank A^T = r
d) Nullity A = r, Rank A = 0, Rank A^T = r

Answer 1

The nullity of matrix A is n rank A, rank A is the dimension of the column space of A, and rank A^T is equal to the rank of A. The correct answer is Option a) Nullity A = r, Rank A = r, Rank A^T = r.

Step-by-step explanation:

The nullity of matrix A is the dimension of the null space of A, which can be calculated using the formula nullity A = n minus rank A, where n is the number of columns in A. Since rank A = r, nullity A = n minus r. The rank of matrix A is the dimension of the column space of A. It represents the maximum number of linearly independent columns in A. Therefore, rank A = r. The rank of matrix A transpose is equal to the rank of A.

Hence, rank A^T = r. Therefore, the correct answer is Option a) Nullity A = r, Rank A = r, Rank A^T = r. If matrix A has rank r, the nullity of A refers to the dimension of the null space of A, which is the set of solutions to the homogeneous system Ax = 0. According to the rank-nullity theorem, the sum of the rank and nullity of matrix A is equal to the number of columns in A. Therefore, if the rank of A is r and A has n columns, then the nullity of A would be n minus r. Also, the rank of a matrix is equal to the rank of its transpose, so the rank of A and AT (transpose) are both r.