# Give example of matrices such that AB=AC but B=/=C

1. Let M(2,R) be the set of all 2 x 2 matrics over R. Give an example of matrices A,B,C in M(2,R) such that AB=AC, but B is not equal to C.

3.

Mark44
Mentor

1. Let M(2,R) be the set of all 2 x 2 matrics over R. Give an example of matrices A,B,C in M(2,R) such that AB=AC, but B is not equal to C.

3.
What have you tried? You have to show some effort before we can provide any help.

What have you tried? You have to show some effort before we can provide any help.

(A^-1)AB=(A^-1)AC so B=C. This shows that A must have no inverse element. So A could be
1 0
0 0
because det(a)=1-0=0 so A has no inverse. I dont know what A and B could be.

Mark44
Mentor

(A^-1)AB=(A^-1)AC so B=C. This shows that A must have no inverse element.
No, what this shows is that if A is invertible (has an inverse), then AB = AC implies that B = C. But you're not given that A is invertible.
So A could be
1 0
0 0
because det(a)=1-0=0 so A has no inverse. I dont know what A and B could be.
You mean B and C. See if you can cobble up different matrices B and C so that AB = 0 and AC = 0, but B != C.

No, what this shows is that if A is invertible (has an inverse), then AB = AC implies that B = C. But you're not given that A is invertible.

You mean B and C. See if you can cobble up different matrices B and C so that AB = 0 and AC = 0, but B != C.

what does AB = 0 mean? does it mean det(A) x det(B) or matrix A x matrix B?

Mark44
Mentor

By 0 I meant the 2 x 2 matrix whose entries are all 0.

By 0 I meant the 2 x 2 matrix whose entries are all 0.

so they could be B = 0 0 and C = 0 0 ??
............................ 0 1.............1 0

Mark44
Mentor

Sure, why not? All you had to do was come up with three 2 x 2 matrices such that AB = AC, but B != C. It looks like you did just what you are asked to do.