Linear Algerbra EigenValues

[Zahara2112]

hey guys i ahve this question that is due in like half an hour and i was wondering if you could help me out?

the question is

For which value of k does the matrix
A=6 k
-5 8

have one real eigenvalue of multiplicity 2?

ive done lots of work on it and basically im just wondering how to do it?

thanks for the help in advance

 

[galactus]

Find the characteristic polynomial. \(\displaystyle Det(A-I{\lambda})\)

It is \(\displaystyle {\lambda}^{2}-14{\lambda}+5k+48=0\)

Now, this is a quadratic. It has a root of multiplicity 2 when its discrimiant is 0.

That is, \(\displaystyle b^{2}-4ac=0\), where \(\displaystyle a=1, \;\ b=-14, \;\ c=48+5k\)

Solve for k.