Find the characteristic polynomial and the eigenvalues of the matrix. [5 3]
[-4 4]

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;

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

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f(x) = - 2/3x + 5; f(5/2)

substitute 5/2 in for x:

f(5/2) = -2/3(5/2) + 5

simplify the right side:

f(5/2) = -5/3 + 5

f(5/2) = 10/3

answer:

f(5/2) = 10/3

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