 # 1. The following is the row-reduced echelon form of a 3 x 3 matrix ...

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1. The following is the row-reduced echelon form of a 3 x 3 matrix A: RREF(A) 000 Note that the following prompts pertain to A. not the result of row-reducing A. Circle one option for each part. You do not need to show your work. (a) (2 points) Ais invertible not invertible / not enough information (b) (2 points) For is an eigenvalue is not an eigenvalue / not enough information (c) (2 points) For A.A=0 is an eigenvalue is not an eigenvalue enough information (d) (2 points) The dimension of the null space of A, i.e., dim{r R3 Ax 0}, is 1 / 2 / not enough information (e) (2 points) For the system A the vector is solution / is not a solution / not enough information (i) (2 points) For the system A s olution exists unique / exists & need not be unique / not enough information (g) (2 points) For the system A given 3(0) o solution [=(0)_ is unique / not unique / not enough information (h) (2 points) The determinant of A²is / nonzero / not enough information (i) (2 points) The first and second columns of are linearly dependent / linearly independent / not enough information 2. (8 points) Give the general solution to the following differential equation: (the following may or may not be useful to 3. (a) (4 points) Give the general solution to 51" 21/" +2y = 0. (b) (6 points) Solve the nonhomogeneous IVP: 2v f, y(0) 0. y'(0) y"(0) 0. 4. Consider the following linear system: x-|17% (a) (6 points) Give the general solution to the linear system. (b) (6 points) Sketch the phase portrait for the linear system. Include at least two characteristic trajectories. 5. (8 points) Given the differential equation y(4) 0. determine if the set of solutions forms basis for the solution space. Be sure to include your reasoning. 6. Consider the first order linear system For each of the following values for k. provide characteristic phase portrait that illustrates how solutions behave as (a) points) *<0 (b) points) o (c) points) k>0 7. (10 points) Give the general solution to the following first order linear system: (hint: the 3x3 theory is similar to the 2x2 theory) X x. 8. Consider the IVP y(1)=0. (a) (4 points) Does solution exist? Be sure to include your reasoning (b) (4 points) If solution exists is it unique? If no solution exists. provide different initial data y(to) 90 such that the resulting IVP has unique solution. Be sure to include your reasoning 9. (a) (4 points) Let A be an invertible matrix with eigenvector and corresponding eigenvalue & Show that A- has the same as eigenvector with corresponding eigenvalue 1/A. (b) (4 points) Let M have eigenvector w with corresponding eigenvalue . Show that for p(x) bz the matrix p(M) aM² bM el has was eigenvector with corresponding eigenvalue P(A) au bu+c. 10. Consider the following nonlinear system of differential equations: y=1-e. (a) (4 points) Find the horizontal and vertical nullclines of the system. (b) (2 points) Find all equilibrium points of the system.

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