Given that v1 = [10] [65 -210] and v2 = are eigenvectors of the matrix 3 21 -68 determine the corresponding eigenvalues. λι = help (numbers) X2 = help (numbers) Find the solution to the linear system of differential equations x' = 65x-210y y' = 21x - 68y satisfying the initial conditions x (0) = 24 and y(0) = 7. x(t) = help (formulas) y(t) = help (formulas) Book: Section 3.4 of Notes on Diffy Qs

The matrix v₁ = 1 x₁ (t) = -13 12] -8 7 x₂ (t) = has eigenvalues A₁ = -1 and ₂ = -5. Find eigenvectors corresponding to these eigenvalues. satisfying the initial conditions 1 2 Find the solution to the linear system of differential equations and 7₂ x₁ (0)] _x₂ (0) = [3] help (formulas) help (formulas) = x 3 x = - 13 -8 12 7 X1 X help (matrices)

Find the eigenvalues and eigenvectors for the coefficient matrix.

Find the solutions of the system of linear equations if the eigenvalues of the matrix A are

Consider the initial value problem -6 70) = -6 a. Find the eigenvalue A, an eigenvector 1, and a generalized eigenvector v, for the coefficient matrix of this linear system. 1 -6 1 b. Find the most general real-valued solution to the linear system of differential equations. Use t as the independent variable in your answers. e^(-6t) (t+1)e^(-6t) ý(t) = c1 + c2 e^(-6t) c. Solve the original initial value problem. Y1(t) 2(t) = e^(-6t)

Suppose a system of differential equations yields eigenvalue λ=1+3i with 2+i eigenvector = Find any real-valued solution.

The eigenvalue and

Solve the system of differential equations{x'=29x−50yy'=15x−26y{x′=29x-50yy′=15x-26yx(0)=−13,  y(0)=−8x(0)=-13,  y(0)=-8The lesser of the two eigenvalues isIts corresponding eigevector is ⟨a,−3⟩⟨a,-3⟩. What is aa?a=a=The greater of the two eigenvalues isIts corresponding eigevector is ⟨2,b⟩⟨2,b⟩. What is bb?b=b=The solution to the system isx(t)x(t) =   y(t)y(t) =

Suppose that a 4 x 4 matrix A has eigenvalues A₁ = 1, A2=-3, A3=5, and A4 = -5. Use the following method to find det (A). If So, on setting λ = 0, we obtain that det (A) = i + Cn p(A) = det (Al-A) = \" + C₁^^¯¹ +... det (-A) = C, or det (A) = (-1)^cn

Find the eigenvenctors to this matrix