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1. (a) Find the eigenvalues ($lambda_1$ and $lambda_2$) and eigenvectors ($e_1$ and $e_2$) of the following matrix: $egin{aligned} A = egin{bmatrix} 8 & 3\ 3 & 0 end{bmatrix} end{aligned}$ (b) Verify that the product of your eigenvalues is the determinant of the matrix. (c) Verify that the sum of your eigenvalues is the trace of the matrix. (d) Verify that $Ae_i = lambda_i e_i$ for $i = 1, 2$. (e) Are your eigenvectors orthogonal? That is, does $e_1^T e_2 = 0$? Did you expect them to be? (f) Normalize your eigenvectors to get the vectors $ ilde{e}_1$ and $ ilde{e}_2$. Form $X_s = [ ilde{e}_1 ilde{e}_2]$. Find $X_s^{-1}$ (Hint: this should be a simple operation). 2. Find the particular solution to the initial value problem (IVP) given below: $egin{aligned} 2 frac{dy(t)}{dt} + 5y(t) = 0, quad y(0) = 50 end{aligned}$ (Recall that you can check your solution by: (i) substituting it back into the differential equation to make sure it works; and, (ii) verifying the initial condition.)

          1. (a) Find the eigenvalues ($lambda_1$ and $lambda_2$) and eigenvectors ($e_1$ and $e_2$) of the following matrix:
$egin{aligned} A = egin{bmatrix} 8 & 3\ 3 & 0 end{bmatrix} end{aligned}$
(b) Verify that the product of your eigenvalues is the determinant of the matrix.
(c) Verify that the sum of your eigenvalues is the trace of the matrix.
(d) Verify that $Ae_i = lambda_i e_i$ for $i = 1, 2$.
(e) Are your eigenvectors orthogonal? That is, does $e_1^T e_2 = 0$? Did you expect them to be?
(f) Normalize your eigenvectors to get the vectors $	ilde{e}_1$ and $	ilde{e}_2$. Form $X_s = [	ilde{e}_1 	ilde{e}_2]$. Find $X_s^{-1}$ (Hint: this should be a simple operation).
2. Find the particular solution to the initial value problem (IVP) given below:
$egin{aligned} 2 frac{dy(t)}{dt} + 5y(t) = 0, quad y(0) = 50 end{aligned}$
(Recall that you can check your solution by: (i) substituting it back into the differential equation to make sure it works; and, (ii) verifying the initial condition.)

$1. (a) Find the eigenvalues (lambda1 and lambda2) and eigenvectors (e1 and e2) of the following matrix: eginaligned A = eginbmatrix 8 3 3 0 endbmatrix endaligned (b) Verify that the product of your eigenvalues is the determinant of the matrix. (c) Verify that the sum of your eigenvalues is the trace of the matrix. (d) Verify that Aei = lambdai ei for i = 1, 2. (e) Are your eigenvectors orthogonal? That is, does e1^T e2 = 0? Did you expect them to be? (f) Normalize your eigenvectors to get the vectors ildee1 and ildee2. Form Xs = [ ildee1 ildee2]. Find Xs^-1 (Hint: this should be a simple operation). 2. Find the particular solution to the initial value problem (IVP) given below: eginaligned 2 fracdy(t)dt + 5y(t) = 0, quad y(0) = 50 endaligned (Recall that you can check your solution by: (i) substituting it back into the differential equation to make sure it works; and, (ii) verifying the initial condition.)$

Added by Irene T.

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