The discussion centers around finding examples of 2 x 2 matrices A, B, and C such that the product of A with B equals the product of A with C (AB = AC), while B and C are not equal. This is a problem within the subject area of linear algebra, specifically dealing with matrix multiplication and properties of matrices.
The discussion is ongoing, with participants exploring various interpretations of the problem. Some have suggested specific forms for matrices A, B, and C, while others are questioning the assumptions about matrix A's invertibility. There is no explicit consensus, but productive lines of reasoning are being explored.
Participants note that A must not be invertible for the condition AB = AC to hold while B and C remain distinct. The discussion also touches on the definition of the zero matrix and its implications in the context of the problem.
What have you tried? You have to show some effort before we can provide any help.xlalcciax said: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 said:What have you tried? You have to show some effort before we can provide any help.
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.xlalcciax said:(A^-1)AB=(A^-1)AC so B=C. This shows that A must have no inverse element.
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 said: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 said: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.
Mark44 said:By 0 I meant the 2 x 2 matrix whose entries are all 0.