How can you prove that if AB = AC and A is nonsingular, then B = C?

[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

 

[HallsofIvy]

What does your textbook say "nonsingular" means for a matrix? What does your textbook say about "invertible"?

 

[hkus10]

What does your textbook say "nonsingular" means for a matrix? What does your textbook say about "invertible"?

An n x n matrix A is called nonsingular, or invertible, if there exists an n x n matrix B such that AB = BA = In; such a B is called inverse of A.

 

[Dick]

An n x n matrix A is called nonsingular, or invertible, if there exists an n x n matrix B such that AB = BA = In; such a B is called inverse of A.

Ok, use that. You usually write the inverse of A as A^(-1). Multiply both sides of AB=AC by A^(-1).