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

the quotient is 4x^2 - 27x + 167 with a remainder of -996

step-by-step explanation:

in synthetic division, use -6 as the divisor and use the coefficients {4, -3, 5, 6}:

-6     )   4     -3     5     6

                -24   162 -1002

          4   -27   167   -996

thus, the quotient is 4x^2 - 27x + 167 with a remainder of -996.

note:   your "4x°-3x+5x+6" is ambiguous / incomplete.   i had to assume that you meant '4x^3 - 3x^2 + 5x + 6."

$450 is the answer.

step-by-step explanation:

30% of 450 = 135

450-135=315