1. The problem statement, all variables and given/known data Let A be an n x n matrix and let x and y be vectors in R^n. Show that if Ax = Ay and x [tex]\neq[/tex] y, then the matrix A must be singular. 2. Relevant equations So far we have learned the definition of a matrix that has an inverse to be one where: if there exists a matrix B and AB = BA = I. The matrix B is said to be the multiplicative inverse of A. 3. The attempt at a solution I have done earlier problems that involved proving things have an inverse with the above definition, however I can not think of how to apply it to this problem. So what I did was use my knowledge from earlier classes (also this concept was touched upon a few problems earlier, but has not yet been defined), that if the determinant = 0 then the matrix does not have an inverse. But I have not found anything that helps me. A = |a11 a12| x = x1 y = y1 |a21 a22| x2 y2 Ax = Ay (this is supposed to read as Ax = Ay expanded, sorry. |a11*x1 + a12*x2| |a11*x1 + a12*x2| | | = | | |a21*x1 + a22*x2| |a21*x1 + a22*x2| I've been getting it into equations like: a11(x1-y1) + a12(x2-y2) = 0, and a few similar things, but I have not yet found something that will fit into the formula for the determinant. So am I on the right track? If i am please give me some hints on how to proceed, if not then let me know what to do please. Thanks a lot, and sorry if anything is unclear.