find the characteristic roots of matrix 3*3 matrix

Question

A= [25  1  2, 1  3  0,  2  0  -4]

 

find the characteristics value of the above matrix

Answer 1

Need to evaluate the determinant for A-xI where x is a scalar and I the identity matrix. The resultant determinant is

| 25-x    1   2 |

|      1 3-x   0 | = 0

|    2    0 -4-x |

produces the characteristic equation where x gives the Eigenvalues.

(25-x)(3-x)(-4-x)-1(-4-x)+2(-(3-x)2)=

-(75-28x+x²)(4+x)+4+x-12+4x=

-(300-112x+4x²+75x-28x²+x³)-8+5x=

-308+24x²+42x-x³=0, or x³-24x²-42x+308=0.

The Eigenvalues are roughly -4, 3, 25, and we can use Newton’s Method to get more accurate values:

-4.13794, 2.95579, 25.18215