These are problems from the text that were advised to study for the next exam. (see attachment for equations)
1) Determine the values of r for which det(A-rI) = 0
2) Verify that X(t) is a fundamental matrix for the given system and compute X-1(t). Use the result, x' = Ax, x(t0) = x0 to find the solution to the given initial value problem.
3) Using matrix algebra techniques, find a general solution of the system.
4) Find the solution to the given system that satisfies the given initial condition.
5) Find the solution to the given system that satisfies the given initial condition.© BrainMass Inc. brainmass.com September 19, 2018, 9:17 am ad1c9bdddf - https://brainmass.com/math/matrices/matrices-values-determinants-92017
Please see the attachment
1. We set
Thus we get or
2. First, we have
Thus we get . So is a fundamental matrix for the given system.
From the fundamental matrix, the general solution of the system is
This shows how to verify a fundamental matrix, find solutions using matrices, and find solutions that satisfy given initial conditions.