Question

Find the characteristic polynomial and the eigenvalues of the matrix.

Answer 1

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;

$= \left[\begin{array}{cc}5&3\\-4&4\\\end{array}\right] - \lambda \left[\begin{array}{cc}1&0\\0&1\\\end{array}\right] = 0\\= \left[\begin{array}{cc}5&3\\-4&4\\\end{array}\right] - \left[\begin{array}{cc}\lambda &0\\0&\lambda \\\end{array}\right] = 0\\= \left[\begin{array}{cc}5-\lambda&3\\-4&4-\lambda\\\end{array}\right] = 0$

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