# Eigenvalues, Eigenvectors. DUE TOMORROW

#### arvis2000

And this as well:

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#### Idea

A4 (a) False. $$\displaystyle A=B=-I$$

A4 (b) True. use the definition. $$\displaystyle Av=a v$$ and $$\displaystyle Bv=b v$$

topsquark

#### Archie

The product of the eigenvectors of a matrix is equal to the determinant. And the product of the determinants of two matrices is equal to the determinant of the product of the matrices. So, if the 2x2 matrix $A$ has eigenvalues $x$ and $y$ and the 2x2 matrix $B$ also has eigenvalues $x$ and $y$, we have $\det{A}=\det{B}=xy$ and thus $\det{AB}=x^2y^2$. But by statement (a) we also have that $x$ and $y$ are the two eigenvectors of $AB$ and so $\det{AB}=xy$.

For (b), think about the transformations that A and B represent and thus determine the transformation that the product AB represents.

topsquark