Let A and B be n × p matrix respectively and p × q matrix and defineC = AB. Prove the following statements.(a) If the columns of B are linearly dependent, then the columns of C are linearlydependent.(b) If the rows of A are linearly dependent, then the rows of C are linearly depen-dent.

Matrix A has full rank ifa.determinant of A is positiveb.rows of Matrix A are linearly dependentc.determinant of A is negatived.columns of Matrix A are linearly dependent

3. (a) If X and Y are subsets of a vector space, and either X or Y is linearly dependent, show thatX ∪ Y is linearly dependent. [6 marks](b) Give an example of a vector space V with subsets X and Y , such that X and Y are linearlyindependent, but X ∪ Y is linearly dependent.

Two matrices A and B are multiplied to get AB if:Question 29Answera.No. of rows of A is equal to no. of columns of B.b.None of these.c.Both are rectangular.d.Both have same order.

Use the fact that matrices A and B are row-equivalent.A = −2 −5 8 0 −17 1 3 −5 1 5−5 −9 13 7 −671 7 −13 5 −3B = 1 0 1 0 1 0 1 −2 0 30 0 0 1 −50 0 0 0 0(a) Find the rank and nullity of A.rank nullity     (b) Find a basis for the nullspace of A. (c) Find a basis for the row space of A. (d) Find a basis for the column space of A. (e) Determine whether or not the rows of A are linearly independent.independentdependent    (f) Let the columns of A be denoted by a1, a2, a3, a4, and a5. Which of the following sets is (are) linearly independent? (Select all that apply.){a1, a2, a4}{a1, a2, a3}{a1, a3, a5}

Which of the following statements is false about the given vectors?a = [1 2 3]b = [2 4 6]c = [0 1 0]d = [0 0 1]e = [4 5 6]f = [5 7 9]Select an option Clear ResponseVector a and b are linearly dependent.Vector f is a linear combination of vectors a and e.Vector a and e are linearly independent.Vector b and c are linearly dependent.