Algebraic Properties of Matrix Operations

[EV33]

1. Homework Statement

Let A and B be (2x2) matrices such that A^2 = AB and A does not equal the zero matrix O. Find the flaw in the following proof that A = B:

Since A^2 = AB, A^2 - AB = the zero matrix O
Factoring yields A(A-B) = O
Since A does not equal O, it follows that A - B = O.
Therefore, A = B.



3. The Attempt at a Solution

I tried setting up two matrices A and B where A = [ a b, c d] and B = [ e f, g h] and following through on the steps of the proof to see if each of the statements was true. However, I kept finding that they were all true.

Please help.
Thanks.

 

[Mark44]

Problem is, the conclusion is not true for all matrices A and B, even when neither is the zero matrix.

Try playing with matrices that have mostly (but not all) zero entries.

 

[HallsofIvy]

1. Homework Statement

Let A and B be (2x2) matrices such that A^2 = AB and A does not equal the zero matrix O. Find the flaw in the following proof that A = B:

Since A^2 = AB, A^2 - AB = the zero matrix O
Factoring yields A(A-B) = O
Since A does not equal O, it follows that A - B = O.

This is not true. The fact that a product of matrices is 0 does NOT imply one of the factors must be 0.
For example
$$\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & 0 \\ 1 & 0\end{bmatrix}= \begin{bmatrix}0 & 0 \\ 0 & 0\end{bmatrix}$$
the 0 matrix.

Therefore, A = B.



3. The Attempt at a Solution

I tried setting up two matrices A and B where A = [ a b, c d] and B = [ e f, g h] and following through on the steps of the proof to see if each of the statements was true. However, I kept finding that they were all true.

Please help.
Thanks.