Math 4133 - Midterm - Mathematica Portion

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1. Consider the matrix A = (({{-1-λ, 1, 0, 0}, {0, -1-λ, -1, 0}, {0, 1, 1-λ, 0}, {0, 0, 1, 1-λ}})).  Find all values of λ such that the equation A X = 0 has a nontrivial solution.

2. For each of the values of λ from problem 1, find all corresponding nontrivial solutions X to A X = 0.

3. Consider the matrix C = (({{2, 2, 2, -3}, {6, 1, 1, -4}, {1, 6, 1, -4}, {1, 1, 6, -4}})).  Find a formula for (C^n), for n≥ 1.

4. Solve for the variable (Subscript[x, 4]) only in the following linear system: (({{5, 2, -3, 2, 1, 5}, {2, -3, 5, 2, 4, 0}, {-1, 3, 7, 0, 0, 2}, {4, 1, 0, -5, 3, 4}, {0, 2, 0, 5, -2, 4}, {2, 0, -3, -1, -1, 1}}))(({{(Subscript[x, 1])}, {(Subscript[x, 2])}, {(Subscript[x, 3])}, {(Subscript[x, 4])}, {(Subscript[x, 5])}, {(Subscript[x, 6])}})) = (({{2}, {1}, {0}, {0}, {-1}, {7}})).  Note: You are not allowed to solve for any other variables in this problem.

5. Find the angles (Subscript[θ, 1]), (Subscript[θ, 2]), (Subscript[θ, 3]) and (Subscript[θ, 4]), that the vector < 2, -1, 3, 4 > makes with (Subscript[x, 1]), (Subscript[x, 2]), (Subscript[x, 3]) and (Subscript[x, 4]) axes in (R^4), respectively.

6. Find the side lengths, the angle between each side, and the area of the triangle, defined by the two vectors <-1, 3, 2, 3, 5> and <2, 3, 1, 1, 1>

7. From problems 20-21 on the written portion, graph the vectors u, v and w, the plane from problem 20, and the vector projection from problem 21.  Also include w-projection in this graph.