How Do You Show |λ|^2 = 1 for a 2D Transformation Matrix?

[UrbanXrisis]

Show by direct expansion that $$| \lambda | ^2 =1$$

For simplicity, take $$\lambda$$ to be a two-dimensional transformation matrix.

from what I understand, if X was a vector (2,3,4), $$| X |$$ is finding the length of the vector by adding the square of the numbers and taking a square root. $$\sqrt{2^2+3^2+4^2}$$

What I don't understand is how to apply this to a matrix

because a 2x2 matrix times itself is still a 2x2 matrix, and even after one square root's it, it's still a 2x2 matrix, never just 1.

What am I missing?

 

[HallsofIvy]

You are missing just about everything! What do you mean "For simplicity, take $\lambda$ to be a two-dimensional transformation matrix"? Is that given as part of the problem? Why "for simplicity"? If you are not told what $\lambda$ is, the problem makes no sense at all.

Exactly what is a "transformation matrix"? You can't mean what I would think it means because it simply is not true that the determinant of every transformation matrix is 1. And it would be a really good idea to look up "determinant of a matrix". If you were asked to do this problem, then you were certainly expected to know what that is and how to calculate it!

 

[UrbanXrisis]

The question just asks

"Show by direct expansion that $$| \lambda | ^2 =1$$ For simplicity, take $$\lambda$$ to be a two-dimensional transformation matrix."

your guess is as good as mine as to what is a transformation matrix.
And it's not "the determinant of every transformation matrix is 1" it's the determinant squared is equal to one, which also doesn't make sense because I thought that $$1=| \lambda | |\lambda|^{-1}$$

 

[UrbanXrisis]

is this a possible description:

$$1= | \lambda | ^2 =| \lambda | |\lambda|^{-1}$$

I'm not really sure on this...