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

[xlalcciax]

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]

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.

 

[xlalcciax]

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 don't know what A and B could be.

 

[Mark44]

(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 don't 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.

 

[xlalcciax]

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]

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

 

[xlalcciax]

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]

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.