Linear Algebra - Find Matrix that fits the criteria

[diogomgf]

Hello every one, first post here on the forum...

I have been struggling with this problem for a while:

Find a matrix A_2x2 such that A² = I​₂.
A can't be neither I₂ or -I₂.Thank you.

 

[tkhunny]

Hello every one, first post here on the forum...

I have been struggling with this problem for a while:

Find a matrix A_2x2 such that A² = I​₂.
A can't be neither I₂ or -I₂.Thank you.

If all else fails, you can Brute Force it.

\(\displaystyle A = \begin{pmatrix}a & b \\ c & d\end{pmatrix}\)


What do you get for \(\displaystyle A^2\;then\;A^{2} = I_{2}\)? Does this suggest anything?

You told us we can't have \(\displaystyle a =1,\;b=0,\;c=0,\;d=1\), so pick something else. To be more general, start small, meaning Don't try to find all four values at the same time. I started with \(\displaystyle c = 1\). This leads to something interesting about the choice of b. There are many possible choices. Do a, b, c, or d have to be Real?

P.S. Eigenvalues are almost magic. You may wish to look into that, too. See where your exploration leads you.

 

[diogomgf]

P.S. Eigenvalues are almost magic. You may wish to look into that, too. See where your exploration leads you.

Thanks alot for the reply, I will have a look into Eigenvalues. Never heard of them!

By brute forcing do you mean to try and solve the polynomial equation system?

Thanks