# Homework Help: Linear Algebra: Idemponent matrix

1. Sep 14, 2009

### Amy-Lee

In idemponent matrices:
Is there a value of r for which
(matrix) A= -1 -1
2 r

is idempotent? If so, what is the value of r and why.

I know that A^2=A, that's about all... How does one find r, if this is the only info given?

I have no idea how to approach this. Please help.

Last edited: Sep 14, 2009
2. Sep 14, 2009

### CFDFEAGURU

What is an indemponent matrix?

Thanks
Matt

3. Sep 14, 2009

### Amy-Lee

If A is an n x n matrix, then A is called idempotent if A^2=A

4. Sep 14, 2009

### CFDFEAGURU

Sorry, can't help you with that. But thanks for letting me no what idempotent means.

Thanks
Matt

5. Sep 14, 2009

### gabbagabbahey

The simple definition is simply a matrix for which $A^2=A$.

Well, if $$A=\begin{pmatrix}-1 & 1 \\ 2 &r \end{pmatrix}$$, what will $A^2$ be?....Setting $A^2=A$ should give you 4 equations (one for each component of the matrices), is there a value of $r$ that solves all 4 equations simultaneously?

6. Sep 14, 2009

### Amy-Lee

A^2 = -1 -1
2 r

and -1 -1 = -1 -1
2 r 2 r ???

7. Sep 14, 2009

### Amy-Lee

sorry for bad layout... still haven't figured out how to use advanced reply

8. Sep 14, 2009

### Preno

Think about the determinant of an idempotent matrix.

9. Sep 14, 2009

### gabbagabbahey

No,

$$A=\begin{pmatrix}-1 & -1 \\ 2 &r \end{pmatrix}\implies A^2=\begin{pmatrix}-1 & -1 \\ 2 &r \end{pmatrix}\begin{pmatrix}-1 & -1 \\ 2 &r \end{pmatrix}$$

Carry out the matrix multiplication.

10. Sep 14, 2009

### Amy-Lee

ok I get:
-1 1-r
-2+2r -2+r2

11. Sep 14, 2009

### gabbagabbahey

Right, and for $A$ to be idempotent, that must also be equal to the matrix $A$...so you want to find an $r$ such that

$$\begin{pmatrix}-1 & 1-r \\ -2+2r &-2+r^2 \end{pmatrix}=\begin{pmatrix}-1 & -1 \\ 2 &r \end{pmatrix}$$

12. Sep 14, 2009

### Amy-Lee

ok thank you... do I now have to reduce it to row-echelon form, by using the Gaussian Elimination?

13. Sep 14, 2009

### HallsofIvy

You don't have to. If $A^2= A$ then you must have -1= -1, -1= 1-r, -2+2r= 2 and -2+r^2= r. Are there values of r that satisfy all of those and, if so, what are they?

14. Sep 14, 2009

### Amy-Lee

r= 2; r=1/2; r=-1

15. Sep 14, 2009

### gabbagabbahey

r=2 works, but the other two values do not satisfy all 4 equations. (For the two matrices to be equal, all 4 of there components must be equal)

16. Sep 15, 2009

### Amy-Lee

thank you so much for your help. One last question on the topic: is a square invertible idempotent matrix also the Identity matrix?

17. Sep 15, 2009

### gabbagabbahey

Well, if $A^2=A$, what can you say about $\detA$?

18. Sep 15, 2009

### Amy-Lee

I don't understand the last part of what you said

19. Sep 15, 2009

### gabbagabbahey

Take the determinant of both sides od the equation $A^2=A$....what do you get?