Question

[3 points) Suppose that A is an m x 3 matrix and that the nullity(A)=2. (a) What is the rank(AT)? (b) If Ax = b is consistent, then the solutions form which of the following? Circle one: Point in R3 Line in R3 Plane in R3 Point in R Line in R Plane in R™ (c) If mullity(AN) = 3, find m.

Answer 1

(a). As per the dimension theorem, rank(A) = no. of columns in A -nullity (A) = 3-2 = 1. Further, since the column rank and row rank are same, hence rank(A^T) = 1. (The rank of a matrix is equal to the rank of its transpose).

(b). If m ≠ 3, and if the equation AX = b is consistent, then this equation has infinite solutions. Further, such solutions are vectors in R³. Therefore, the solution set is a plane in R³ ( being the span of vectors in R³). The 3^rd option is the correct answer.

However, if m = 3, then there could be a unique solution to AX = b if A is invertible. Then the solution to AX = b could be a point in R³ .

( c). Since A^T is a 3 x m matrix, If nullity(A^T) = 3, hence as per the dimension theorem, rank(A^T) = m-3.

Further, since the rank of a matrix is equal to the rank of its transpose and since A has only 3 columns, hence rank(A) ≤ 3 i.e. m-3 ≤ 3 so that m ≤ 6.