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

Question 1099338: Please help me
Give examples to show that the following statements are wrong.
1) if (Matrix) , 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(52909) (Show Source): You can put this solution on YOUR website!
.
Give examples to show that the following statements are wrong.
1) if (Matrix) , 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 = .


    Next easiest counter-example:  A = .


    Matrices with the property   = 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 = ,  X = ,  Y = .   (where X and Y are vectors)


    Next easiest counter-example:  A = ,  X = ,  Y = .


    You make all necessary calculations . . .

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