# Theoretical Question

• Oct 27th 2009, 07:40 PM
satweety12
Theoretical Question
We have a real 2x2 matrix A with 2 distinct real eigenvalues, lambda1 and lambda 2. explain the formula detA = lambda1(lambda 2) geometrically, thinking of module(detA) as an explansion facor. illustrate your explanation with a sketch. can you explain this similarly for 3x3 matrices?

i dont even know where to begin! and thanks for the help in advance!
• Oct 27th 2009, 07:58 PM
tonio
Quote:

Originally Posted by satweety12
We have a real 2x2 matrix A with 2 distinct real eigenvalues, lambda1 and lambda 2. explain the formula detA = lambda1(lambda 2) geometrically, thinking of module(detA) as an explansion facor. illustrate your explanation with a sketch. can you explain this similarly for 3x3 matrices?

i dont even know where to begin! and thanks for the help in advance!

I'm not completely sure whether the question is clear enough to me, but I can think of the following: if the matrix $A$ has two eigenvalues $\lambda_1\neq \lambda_2$, then $A$ is similar to the matrix $A'=\left(\begin{array}{cc}\lambda_1&0\\0&\lambda_2 \end{array}\right)$ , and since we know that $\det A=\det A'$ , and this last one has a very nice and simple geometric interpretation we get (?) what we want: it is the "volume" (generalized algebraic volume, which in this case is 2-dimensional volume = usual area) of the parallelogram (rectangle in this case) generated by the 2-dimensional vectors $(\lambda_1\,\;0)\,,\;(0\,,\;\lambda_2)\,\;in\,\;\m athbb{R}^2$

Tonio
• Oct 27th 2009, 08:14 PM
satweety12
thanks!
thanks!