Applied Linear Algebra problem

[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 won't 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 doesn't have a solution for this one

 

[Studiot]

How about

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

nonzero symmetric 2x2 matrix

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

 

[HallsofIvy]

anonymity, yes, your analysis is correct.

 

[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

 

[Stephen Tashi]

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

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

 

[anonymity]

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

Very clever. Thank you ^