How Do Eigenvalues Determine the Value of a Matrix's Determinant?

[deana]

Homework Statement

Let A be the matrix with eigenvalues x1 = 2, x2 = 1, x3 = 1/2 , x4 = 10

and corresponding eigenvectors v1: <1,-1,1,0>, v2: <1,-1,0,0>, v3: <1,0,0,1>, v4: <0,0,1,1>

Calculate |A|

Homework Equations

See above

The Attempt at a Solution

I'm not really sure how to start this problem but i know that:
For nxn matrices X, Y , Z
|XYZ| = |X| |Y| |Z| and |X^ (-1)|= 1 / |X|
Maybe I could use this to solve the problem?

Any input or suggestions about how to start this problem would be helpful!
Thanks!:)

 

[Dick]

What does the matrix of your linear transformation look like if you express it in the basis {v1,v2,v3,v4}?

 

[Ray Vickson]

Do you know the relationship between the eigenvalues of a matrix and the determinant of that matrix? It is a standard result. If it is not in your textbook or course notes, it can certainly be found through Google.

RGV