(a) If A is invertible and AB = AC, prove quickly that B = C. (b) If A-| | find two different matrices such that AB = AC.
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B = C (b) If Bvi,.., Bvh} is a then vi, . ., vk} is a linearly independent set in R". linearly independent set in R* where B is a kx n matrix,
5. Prove or disprove the following statements (a) Let A B and C be 2 x 2 matrices. If AB = AC, then B...
11. Prove one of the following: a. Let A and B be square matrices. If det(AB) + 0, explain why B is invertible. b. Suppose A is an nxn matrix and the equation Ax = 0 has a nontrivial solution. Explain why Rank A<n.
Using congruence axiom 1 prove that If A-B-C then ABAC,
AB>AC or AC>AB
9. a) Prove that ifA is invertible and AB-AC then B = C.
(9) True of False: For all (n xn) matrices A and B, dim(ker(AB)) > dim(ker(B)). (That is, the dimension of the nullspace or kernel of AB is at least as big as the dimension of the nullspace or kernel of B.) Justify your answer. (10) (Extra Credit) Let A be any nxn matrix. If n is odd, prove that it is impossible for im(A) = (A).
Linear Algebra
Please show details. Thank you.
36. Proof Prove that if A and B are similar matrices and A is nonsingular, then B is also nonsingular and A-1 and B-1 are similar matrices.
Prove the following using appropriate methods. 10) a AC + AB + BC = AB + BC + AC b. AC + BC + AB EBC + AB+ AC
PLEASE PROVE PARTS a and b by CONTRADICTION
and solve for c as well! Could you explain your steps as well
2. (a) (10 marks) Suppose A is an n x n real matrix. Show that A can be written as a sum of two invertible matrices. HINT: for any lER, we can write A = XI + (A - XI) (b) (10 marks) Suppose V is a proper subspace of Mnn(R). That is to say, V is a subspace,...