A= [25 1 2, 1 3 0, 2 0 -4]

find the characteristics value of the above matrix

find the characteristics value of the above matrix

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

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