Math 311-101 Additional Exercises — Summer I, 2012
These are not to be handed in.
Row reduction and determinants
- For each matrix below, answer the following
questions.
\[ A =
\left(\begin{array}{cccc}
1 & -1 & 4 & 2\\
-2 & 1 & 0 & 1\\
-1 & 0 & 4 & 3\\
1 & 1 & 6 &3
\end{array}\right) \qquad
B =
\left(\begin{array}{cccc}
1 & 0 & -1 & 0\\
2 & 1 & 0 & 3\\
0& 1 & 5 & 2\\
-1 & 2 & -2 &1
\end{array}\right)
\]
- Find the determinant via the method of row operations.
- Using the determinant of the matrix, state whether the matrix is
invertible. If it is, find its inverse via row reduction; also, find
the determinant of its inverse.
- Is there a nonzero vector $\mathbf x$ such that the matrix times the
vector is $\mathbf 0$.? Why?
- For 4×4 matrices, state the elementary matrix corresponding
to the given row operation.
- $R_3 = R_3+2R_1$
- $R_2\leftrightarrow R_4$
- $R_1=5R_1$
- For the matrix $P$ from the previous problem, use row reduction
methods to find $P^{-1}$, and then find $\det(P^{-1})$ via any
method. Compare your answer with $\frac{1}{\det(P)}$.
- Let $A$ be a 5×5 matrix, and let its columns be ${\mathbf
a}_j$. Suppose that $3{\mathbf a}_1 - 2{\mathbf a}_5=\mathbf 0$. Is
$A$ invertible? Why?
- Let $P=\left(\begin{array}{ccc}1
&2&1\\0&4&1\\1&1&0\end{array}\right)$ and let $Q
=\left(\begin{array}{ccc}1
&1&-1\\1&-1&4\\2&0&-3\end{array}\right)$. Find the determinants of
$P$ and $Q$ using the methods outlined in class (see also p. 111 in
the text). Use these to find $\det(PQ)$.
- Consider the system of equations below. Use Cramer's rule to find
$x_2$.
\[
\begin{gather}
x_1+2x_2-x_3=1\\ 3x_1+x_2=-2\\ 2x_1+x_2-5x_3=0
\end{gather}
\]
Vector spaces
- §3.2 (pp. 141-144): 1(a,d), 5(a,b,c), 9(d).
- §3.3 (pp. 154-156): 4(b,c), 6(a,d), 7(a,c).
- §3.4 (pp. 161-162): 4, 8(a,b,c), 10
- Let $F = [e^x,e^{-x}]$ and let $V = \text{span}(F)$.
- Show that $F$ is a basis for $V$. (Of course, we already know that it spans.)
- Recall that $\cosh(x) := (e^x+e^{-x})/2$ and $\sinh(x) := (e^x-
e^{-x})/2$.
Find the coordinate vectors for $[\cosh(x)]_F$, $[\sinh(x)]_F$, and $[3\cosh(x)+2\sinh(x)]_F$.
- §3.6 (pp. 180-183): 3(a,b), 4(d,e,f), 10.
- Let A be the matrix given below. Find bases for the column
space, row space, and null space of A. What is the rank of A? What is
the nullity of A?
\[
A =
\left(\begin{array}{cccc}
1 & -2 & 3 & 3\\
2 & -5 & 7 & 3\\
-1 & 3 & -4 &3
\end{array}\right)
\]
Updated 6/9/2012 (fjn).