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?

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

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?