Special Type of Matrices

[hkus10]

1) show that if AB = AC and A is nonsingular, then B = C.

 

2) show that if A is nonsingular and AB = 0 for an n x n matrix B, then B = 0.

 

3) Consider the homogenous system Ax=0, where A is n x n. If A is nonsingular, show that the only solution is the trivial one, x=0.

 

4) Prove that if A is symmetric and nonsingular, then A^-1 is symmetric.

 

Please help and show all your work or at least give me some directions!

 

Thanks

[DrRocket]

1) show that if AB = AC and A is nonsingular, then B = C.

 

2) show that if A is nonsingular and AB = 0 for an n x n matrix B, then B = 0.

 

3) Consider the homogenous system Ax=0, where A is n x n. If A is nonsingular, show that the only solution is the trivial one, x=0.

 

4) Prove that if A is symmetric and nonsingular, then A^-1 is symmetric.

 

Please help and show all your work or at least give me some directions!

 

Thanks

 

Hint. None of these should take more than about five steps, and most quite a bit fewer.