Answer:
Step-by-step explanation:
Given the matrix
[5 3]
[-4 4]
The characteristic polynomial is expressed as |A - λI| = 0 where λ are the eigen values and I is am identity matrix A 2*2 identity matrix is expressed as;
[1 0]
[0 1]
Substitute into expression will give;
[tex]= \left[\begin{array}{cc}5&3\\-4&4\\\end{array}\right] - \lambda \left[\begin{array}{cc}1&0\\0&1\\\end{array}\right] = 0\\= \left[\begin{array}{cc}5&3\\-4&4\\\end{array}\right] - \left[\begin{array}{cc}\lambda &0\\0&\lambda \\\end{array}\right] = 0\\= \left[\begin{array}{cc}5-\lambda&3\\-4&4-\lambda\\\end{array}\right] = 0[/tex]
Find the determinant of the resulting matrix;
|A - λI| = (5- λ)(4- λ)- 3(-4) = 0
|A - λI| = 20-5 λ-4 λ+ λ²+12 = 0
|A - λI| = 20-9λ+λ²+12 = 0
-9λ+λ²+32 = 0
Rearrange;
λ²-9λ+32 = 0
Hence the characteristic polynomial is expressed as λ²-9λ+32 = 0
Get the eigen values by finding the roots of the equation;
λ = 9±√9²-4(1)(32)/2
λ = 9±√81-(128)/2
λ = 9±√-47/2
λ = 9±√-47/2
λ₁ = 9-√47 i/2 and λ₂ = 9+√47 i/2
