Proving Linear Algebra Concepts: Rank, RREF, Invertibility, and Dependency

[astronomophosis]

1) Find two matrices A and B where Rank [AB]≠Rank(BA)

2) Find a matrix A where Rref(A)≠Rref(A^T) where T is the transpose

3) Find X given that B is invertible if BXB^-1 –A=I_n (identity matrix)

4) Prove that [Ab_1 Ab_2 Ab_3] is linearly dependent given that {b1 b2 b3} is linearly dependent.

i can't get any of these and tried substituting numbers and nonzero rows and columns to obtain any of the four. Can someone please help me get these? Thank you to those who help in advance!

 

[0rthodontist]

1-2 is just a matter of trying more matrices. 3 is just algebra. For 4 you should use the definition of linear dependence.