University of Toronto
Faculty of Applied Science and Engineering
Department of Chemical Engineering & Applied Chemistry

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MAT198F - Term Test #1 - R. Luus
Linear Algebra - October 11, 2001

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Note:
Please answer all three questions.
Duration of test is 1 hour.
All questions have equal value.
Closed book test - no aids allowed.
No programmable calculators allowed.

1. Given ,
(a) Calculate the eigenvalues of A.
(b) Show that the sum of the eigenvalues is equal to the trace of A, and the product of the eigenvalues is det A.
(c) Evaluate the eigenvectors corresponding to the eigenvalues obtained in (a).
(d) Calculate the angle between any two of the eigenvectors obtained in (c).

2. Consider the surface

(a) Show that the point (2,2,1) is on the surface.
(b) Obtain an equation for the plane (in standard form) that is tangent to the surface at that point, i.e., at the point (2,2,1).
(c) By using a projection of a suitable vector, obtain equations of two planes which are parallel to the plane in (b), but are a distance 3 from it.
(d) Which of the two planes in (c) is closer to the origin (0,0,0)?

3. In the given circuit on the left:

(a) After setting up the appropriate equations, use Cramer's rule to obtain the current .

(b) Use any method (except copying from someone) to obtain the currents and .