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

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  • #1
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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.
 

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  • #2
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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.
 
  • #3
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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.
 
  • #4
34,896
6,638


(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.
 
  • #5
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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?
 
  • #6
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By 0 I meant the 2 x 2 matrix whose entries are all 0.
 
  • #7
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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
 
  • #8
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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.
 

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