SOLUTION: Please help me Give examples to show that the following statements are wrong. 1) if (Matrix) {{{A^2 = A}}}, then A = 0 or A is an Identity matrix 2) if AX = AY, and Y is no

Algebra ->  Matrices-and-determiminant -> SOLUTION: Please help me Give examples to show that the following statements are wrong. 1) if (Matrix) {{{A^2 = A}}}, then A = 0 or A is an Identity matrix 2) if AX = AY, and Y is no      Log On


   

Question 1099338: Please help me
Give examples to show that the following statements are wrong.
1) if (Matrix) A%5E2+=+A, then A = 0 or A is an Identity matrix
2) if AX = AY, and Y is not equal to 0, the X = Y
Answer by ikleyn(52903) About Me  (Show Source):
You can put this solution on YOUR website!
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Give examples to show that the following statements are wrong.
1) if (Matrix) A%5E2+=+A, then A = 0 or A is an Identity matrix
2) if AX = AY, and Y is not equal to 0, the X = Y
~~~~~~~~~~~~~~~~~~~~~~~~ 

a)  Easiest counter-example:       A = %28matrix%282%2C2%2C+1%2C0%2C++1%2C0%29%29.


    Next easiest counter-example:  A = %28matrix%283%2C3%2C+1%2C0%2C0%2C++0%2C0%2C0%2C+0%2C0%2C1%29%29.


    Matrices with the property  A%5E2 = A  are called idempotent.


    For further reading see, for example, this Wikipedia article

               https://en.wikipedia.org/wiki/Idempotent_matrix


b)  Easiest counter-example:       A = %28matrix%282%2C2%2C+1%2C0%2C++0%2C0%29%29,  X = %28matrix%282%2C1%2C+1%2C0%29%29,  Y = %28matrix%282%2C1%2C+1%2C2%29%29.   (where X and Y are vectors)


    Next easiest counter-example:  A = %28matrix%282%2C2%2C+1%2C0%2C++0%2C0%29%29,  X = %28matrix%282%2C2%2C+1%2C0%2C+0%2C0%29%29,  Y = %28matrix%282%2C2%2C+1%2C0%2C+0%2C1%29%29.


    You make all necessary calculations . . .