MAT242-ANSWERS TO SELECTED PROBLEMS

FROM THE REVIEW FOR FINAL



A3.
(c) $AB=\left[\begin{array} {cc}14&300\\ 28&508\\ 1&60\end{array}\right]$,BA not possible.
A4.
(g) $\left[\begin{array} {cc}8&7\\ 6&6\end{array}\right]$.
B2.
(a) $\left[\begin{array} {rrr}-2/3&-2&1\\ 2/3&1&0\\ -1/3&0&1\end{array}\right]$.
B3.
(b) b=-4.
E1.
(c) ${\bf v}_1=2{\bf v}_2+5{\bf v}_3+5{\bf v}_4$.
F2.
(a) $\{{\bf v}_1,{\bf v}_2\}$
G1.
(a) orthonormal vectors: ${\bf u}_1=\frac{1}{2}(1,1,1,1),\; {\bf u}_2=\frac{1}{2}(1,-1,-1,1),\; {\bf u}_3=\frac{1}{2}(1,1,-1,-1)$.
G2.
${\bf u}_1=\frac{1}{\sqrt{11}}(3,1,1,0),\; {\bf u}_2=\frac{1}{\sqrt{187}}(-1,7,-4,-11).$
H1.
Plane spanned by $(0,1,0),\; (1,0,1).$
H2.
Plane spanned by $(1,1,0),\;(1,0,1)$.
I1.
$-\frac{1}{3}(1,-2,1)$.
I3.
$ \frac{1}{2}(0,1,-1).$
J1.
(c) $ {\bf x}=\frac{1}{3}(-2,-1,3)$(d) $ {\bf x}=(0,2,-1/4).$
K1.
(b) No, not enough linearly independent eigenvectors.

K2.
(a) $D=\left[\begin{array} {ccc}6&0&0\\ 0&6&0\\ 0&0&0\end{array}\right]$,$P=\left[\begin{array} {rrr}\frac{1}{\sqrt{30}}& \frac{2}{\sqrt{5}}&\frac{1}{\sq... ...}{\sqrt{6}}\\ [5pt] \frac{5}{\sqrt{30}}&0&-\frac{1}{\sqrt{6}}\end{array}\right]$

Stefania Tracogna
12/11/1997