110.201 LINEAR ALGEBRA 8 October 1997

First Examination

50 minutes. Closed book. No notes. 80 points, 20 per question. Partial credit may be available, but only if you show your working. HINT: Many answers can be checked by direct substitution or other methods. Use only the officially provided blue books. Begin each of the four questions on a fresh page and number it clearly in the

margin.

1. (a) Solve the linear system

\{ \begin{matrix}x & +y & −2z & +t & = & 5 \\ 2x & +y & −3z & −t & = & 0 \\ x & −y & & −t & = & 1 \end{matrix}

(b) Note that the solution in (a) is not unique.

(i) Is there a solution with x = 75? If so, use (a) to write one down. (ii) Is there a solution with y = 75? If so, use (a) to write one down. (iii) Is there a solution with z = 75? If so, use (a) to write one down. (iv) Is there a solution with t = 75? If so, use (a) to write one down.

2. (a) Evaluate the determinant (where t is a variable)

d = | \begin{matrix}1 & 1 & 1 \\ 1 & 3 & 9 \\ t & t^{2} & t^{3} \end{matrix} |

(b) For what values of t is d = 0?

(c) Without using (a), explain why the values of t listed in (b) make d = 0.

3. For each of the statements below, state whether it is true or false, and give a reason. All matrices appearing are understood to be n × n matrices.

(a) (A + B)^{2} = A^{2} + 2AB + B^{2};

(b) (AB)^{−1} = A^{−1}B^{−1}, assuming that A and B are invertible;

(c) 5(A + B) = 5A + 5B;

(d) If A^{2} = I, then A = ±I;

(e) If AB = I, then B = A^{−1}.

4. By any method, compute the inverse of the matrix

[ \begin{matrix}1 & 2 & 3 \\ 1 & 1 & 2 \\ 2 & −3 & 1 \end{matrix} ]

THIS QUESTION SHEET MAY BE RETAINED JMB