Solve the system Solution First find eigenvalues of A: det(A - kI) = 0 ==> (-20-

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First find eigenvalues of A: det(A - kI) = 0 ==> (-20-k)(20-k) - (-48)(8) = 0 ==> k^2 -20k + 20k - 400 + 384 = 0 ==> k^2 = 16 ==> k = 4 or -4 now find eigenvectors for each... k=4: solve (A-4I)x = 0, so reduce A - 4I to get: 3 -1 0 0 so an eigenvevector is [1 3]^T k= -4: solve (A+4I)x = 0, so reduce A+4I to get: 2 -1 0 0 so an eigenvector is [1 2]^T so solution is of form c1 [1 3]^T e^(4t) + c2 [1 2]^T e^(-4t) plugging in initial condition we have: [-3 -7]^T = c1 [1 3]^T + c2 [1 2]^T so c1 + c2 = -3 and 3c1 + 2c1 = -7 adding -2 times first equation to second gives c1 = -1 and thus c2 = 4 so solution is -e^(4t) + 4e^(-4t) 12e^(4t) + 8e^(-4t) where the first function goes in the first blank of the answer and the second in the second.

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