Applied Linear Algebra problem

1. Sep 5, 2011

anonymity

the question:

the matrix

1 -1
1 -1

has the property that A2 = 0. Is it possible for a nonzero symmetric 2x2 matrix to have this property? Prove your answer.

my work:

for a 2x2 matrix A to be its own inverse, it has to have the form

a b
b a

This squared is

(a2 + b 2) (2ab)
(2ab) (a2 + b2)

(things in parenthesis are their own elements -- it wont save the spaces)

Because there are no real numbers so that a2 + b2 = 0, there is no 2x2 symmetric matrix that has its square equal to the zero vector.

edit: ^^ other than a = 0, and b = 0, which would be a 2x2 zero matrix -- something taken to account in the statement of the question

Is this right? My book doesnt have a solution for this one

Last edited: Sep 5, 2011
2. Sep 5, 2011

Studiot

$$\left( {\begin{array}{*{20}{c}} 2 & 4 \\ { - 1} & { - 2} \\ \end{array}} \right)$$

Edit
Sorry I thought you meant a nonsymmetric matrix. What do you mean by this?

Last edited: Sep 5, 2011
3. Sep 5, 2011

HallsofIvy

Staff Emeritus
anonymity, yes, your analysis is correct.

4. Sep 5, 2011

anonymity

How did you write that matrix in physicsforum's latex?!

And by nonzero they just mean it's not

0 0
0 0

Thanks for responding hallsofivy

5. Sep 6, 2011

Stephen Tashi

Click on the "quote" box in hallsofivy's post and look at how he wrote the matrix in your message composer window.

6. Sep 6, 2011

anonymity

Very clever. Thank you ^