Question

Decide if the following statements are true or false. Give a proof to all the true statements and a counterexample to the false ones.

(a) For all A,B∈Rⁿˣⁿ if A is invertible and AB=0 then B=0.

Answer 1

The statement (a) is false. A counterexample is when A is invertible and AB = 0, but B is not equal to 0. The statement (b) is false. A counterexample is when A × F = B × F, but A is not equal to B. The statement (c) is true. If FÃ = BF, then A = B.

Step-by-step explanation:

The statement (a) is false. If A is invertible and AB = 0, we cannot conclude that B = 0. To counterexample this statement, let A = [1 0] and B = [0 0]. A is invertible because its determinant is non-zero, but AB = [0 0] = 0, and B is not equal to 0.

The statement (b) is also false. We cannot conclude that A = B if A × F = B × F. To counterexample this statement, let A = [1 0] and B = [0 1]. A × F = [1F 0F] = [F 0] and B × F = [0F 1F] = [0 F], which are equal, but A is not equal to B.

The statement (c) is true. If Fà = BF, we can conclude that A = B. To prove this, we can multiply both sides by F^(-1) on the right to get F^(-1)Fà = F^(-1)BF. Since F^(-1)F = I (the identity matrix), we have à = B, and therefore A = B.