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...