Linear Algebra HELP!!! math 211/221

[Shift_Perform]

OK, its a friday night and im tired, i want to get this webwork finished, so those who are math wizzes please help out, i have 4 questions...


1.For which value of K does the matrix

1 K
8 -6


have one real eigenvalue of multiplicity 2 ?


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2.
The matrix
-2 0 1
0 -3 0
-1 0 -4

has one real eigenvalue with two eigenvectors. (evalue is -3, but i cant figure out the eigenvectors)

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3.Give an example of a 2x2 matrix without any real eigenvalues


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4.
Let A:
5 0.25
5 7

Find an invertible P and a diagonal D such that A=PDP^(-1)

P= ?
D= ?


Thanks for reading and hope to get some answers, number 3 is probably the easiest but i just cant find one. also hope u guys can understand the matrix, i couldnt put the brackets

[Shift_Perform]

HELP NEED THIS DONE IN LIKE A COUPLE HOURS AND IM DEAD TIRED OF THIS

[TVG]

2.
The matrix
-2-(-3) 0 1 . X1 = 0
0 -3-(-3) 0 . X2 = 0
-1 0 -4-(-3) . X3 = 0


the x's are a matrix that you multiply the original matrix minus the eigenvalue by. It equals 0

From this you will get equations which you solve algebraically.

The one thing to remember when solving them is to normalize the results with something like. X1^2 + X2^2 = 1

By the way the answer is (1,0,-1) and (0,1,0)

[TVG]

I have a 1200 word essay to write for tommorrow, so i cant help with the other ones right now, but for ones where you have to find a matrix, if you do enough googling, you will find a similar question and solution.