(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let A be an n x n matrix and letxandybe vectors in R^n. Show that if Ax= Ayandx[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= x1y= y1

|a21 a22| x2 y2

Ax= Ay

(this is supposed to read as Ax= Ayexpanded, 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.

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# Homework Help: Introductory Linear Alebra proof question

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