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## Homework Statement

(a) Find the characteristic equation and

(b) the eigenvalues and corresponding eigen-vectors

[-5 0 0

3 7 0

4 -2 3]

## Homework Equations

det([tex]\lambda[/tex]*I - A)

## The Attempt at a Solution

Finding the characteristic equation wasn't that challenging, I took lambda, [tex]\lambda[/tex] , and subtracted the given matrix values from it for every diagonal entry. The determiant of a 3x3 matrix is just the product of these diagonals.....I think so my characteristic equation is ([tex]\lambda[/tex] + 5)([tex]\lambda[/tex]-7)([tex]\lambda[/tex] - 3)

Next the eigenvalues are the values for [tex]\lambda[/tex] that when plugged in would make the characteristic equation equal zero.....so the eigenvalues are -5, 7, 3

This is where I'm confused.

To find the eigen-vectors I'm supposed to multiply each eigenvalue by the identity matrix and subtract the original given matrix: so for the eigenvalue of -5....

[-5 0 0

0 -5 0

0 0 -5]

MINUS

[-5 0 0

3 7 0

4 -2 3]

which equals

[0 0 0

-3 -12 0

-4 2 -8]

I took the reduced row echelon form of this matrix and got...

[1 0 1.777

0 1 -.4444

0 0 0]

Is my eigen vector in the matrix above? I am so confused....I also tried to solve the system generated by multiplying a 3x1 vector of variable components x, y, z times

[0 0 0

-3 -12 0

-4 2 -8]

I appear to have failed miserably at this....because the vector I got did not work. Can someone help walk me through this?