Compute det$(\mA)\,$ by expansion along a row or column of your own choice.
hint
Of course, you choose a row or a column with as many 0’s as possible.
answer
$12\,.$
B
Using row operations reduce $\,\mA\,$ to a triangular matrix, and use this to compute det$(\mA)\,.$
hint
See Theorem 9.13 and Theorem 9.16.
answer
$12\,.$
Exercise 2: Determinants and Rank
C
Repetition: Given the polynomial $\,P(x)=-x^6+x^5+x^4-x^3\,.$ Factorize $\,P(x)\,$ by first putting$\,-x^3\,$outside a bracket leaving a third degree polynomial in the bracket. Find the roots in this, and then state all roots in $\,P(x)\,$ together with their algebraic multiplicities.
answer
$x \in \lbrace -1,0,1 \rbrace$.
The root -1 has multiplicity 1, the root 0 has multiplicity 3 and the root 1 has multiplicity 2. The sum of multiplicities is as expected equal to 6.
Given the matrix \begin{equation}
\mA = \begin{matr}{llll} 1 & a & a^2 & a^3 \\ 1 & 0 & a^2 & a^3 \\ 1 & a & a & a^3 \\ 1 & a & a^2 & a \end{matr}, \quad \mathrm{where} \quad a \in \reel.
\end{equation}
D
Determine (readily using Maple) the determinant of $\mA\,.$
answer
det$\mA=-a^6+a^5+a^4-a^3\,.$
E
For which values of $\,a\,$ is $\,\mA\,$ a singular matrix?
answer
$a \in \lbrace -1,0,1 \rbrace$.
F
Find the rank of $\,\mA\,$ for $\,a \in \lbrace -4, -3, -2, -1, 0, 1, 2, 3, 4 \rbrace\,.$
What has the rank to do with the roots in the determinant found above?
answer
For $\,a \in \lbrace -4,-3,-2,2,3,4 \rbrace,$ the rank is 4, i.e. $\,\mA\,$ has full rank. For $\,a \in \lbrace -1,0,1 \rbrace$ the rank is 3,1 and 2, respectively.
%NB: At summen af en rods multiplicitet og matricens rang i denne opgave lig med antallet af søjler i matricen.
G
Find the rank of $\mA$ for all $ a \in \reel $.
answer
For $a \in \reel \setminus \lbrace -1,0,1 \rbrace$ the rank is 4, for
$a \in \lbrace -1,0,1 \rbrace$ the rank is 3, 1 and 2, respectively.
Exercise 3: Tease Exercise in which the Identity Matrix Appears
The following questions are solved by hand and by clever brain work!
Explain using determinants that $\,\mA\,$ and $\,\mB\,$ are regular and thus invertible. Can you from this conclude that $\,\mA\mB\,$ is regular and invertibel?
hint
Is it easy to compute det$(\mA\mB)\,,$ and to what can we use this?
I
Compute $\,\mA\mC\,,$$\,\mB\mD \,$ and $ \mD\mC $.
Compute $\det(\mA) $ and $\det(\mB) $ using Maple.
answer
$\det(\mA)=2$ and $\det(\mB)=31$.
M
Compute $\det(\mA^7)$ og $\det(\mA^{\transp}\mB)$ without using Maple.
answer
$\det(\mA^7)=(\det(\mA))^7=2^7=128$ and $\det(\mA\transp\mB)=\det(\mA)\cdot\det(\mB)=2\cdot31=62$.
N
Show that $\mA$ has an inverse, and state $\det(\mA^{-1})$ and $\det(\mA^{-7})$.
hint
You may possibly use the arithmetic rules for $\det(\mA^{-n})$ in Theorem 9.20.
%\tref{NUID11-tn5.thmDetEgenskaber}{sætning}.
answer
An inverse exists, since $\det(\mA)\neq 0$, so $\mA$ is regular.
$\det(\mA^{-1})=(\det\mA)^{-1}=\frac{1}{\det\mA}=\frac{1}{2}$ and $\det(\mA^{-7})=\frac{1}{(\det\mA)^{7}}=\frac{1}{128}$.
Exercise 5: Vectors: Addition and Multiplication by a Scalar
O
Draw two vectors $\,\mathbf a\,$ and $\,\mathbf b\,$ on a piece of paper. Construct the vectors $\,\mathbf a+\mathbf b\,$ and $\,\mathbf a-\mathbf b\,.$
P
Then you try the product of a vector and a scalar. Draw a vector $\,\mathbf c\,$ on a piece of paper. How does the vectors $\,\frac 12 \mathbf c\,$ an $-3\mathbf c\,$ look like?
Q
Open the GeoGebra-sheet ParametricRepresentation. Construct the following set of points:
\begin{align}
A=&\left{\,P\,|\,\stackrel{\rightarrow}{OP}=\mathbf v+t\mathbf u\,,\,\,t\in \reel\,\right}\
B=&\left{\,P\,|\,\stackrel{\rightarrow}{OP}=\mathbf v+t(\mathbf u-\mathbf v)\,,\,\,t\in \reel\,\right}\\
C=&\left{\,P\,|\,\stackrel{\rightarrow}{OP}=\mathbf v+s\mathbf u+t(\mathbf u-\mathbf v)\,,\,\,s\in \left[0,1\right]\,,\,t\in \left[0,1\right]\,\right}
\end{align}
hint
You may possibly use the tool parallel line and the polygon-tool as you go along.
Exercise 6: Linear Combinations
In the plane the following vectors are given $\mathbf u,\,\mathbf v,\,\mathbf s\,\,\mathrm{and}\,\, \mathbf t$, together with the parallelogram $A$, see the figure.
R
State $\mathbf s$ as a linear combination of $\mathbf u\,\, \mathrm{and}\,\,\mathbf v$.
answer
$\mathbf s=-\mathbf u +\mathbf 2\mv\,.$
S
Show that $\mathbf v$ can be expressed by the linear combination
Express both $\mathbf s$ and $\mathbf t$ as a linear combination of $\mathbf u$ and $\mathbf v$, and substitute the two linear combinations in place of $\mathbf s$ and $\mathbf t$ in the right-hand side of the equation.
T
Determine four real numbers $a,\,b,\,c\,\,\mathrm{and}\,\,d$ such that $A$ can be described by the parametric representation
This exercise includes three different scenarios in the plane, see the figure.
U
Decide for each of the set of vectors $(\mathbf u,\mathbf v)$, $(\mathbf r,\mathbf s)$ and $(\mathbf a,\mathbf b,\mathbf c)$ whether or not they are linearly independent. If not the zero-vector shall be written as a proper linear combination of the vectors in the set.
hint
See Definition 10.20 in eNote 10.
hint
See Theorem 10.24 in eNote 10.
answer
$\mathbf r$ and $\mathbf s$ are not parallel, therefore they are linearly independent.
$\mathbf u$ and $\mathbf v$ are linearly dependent since $\mathbf u=-2 \mathbf v\,$. Thus: $\mathbf u+2\mathbf v=\mnul\,$.
$\mathbf a$, $\mathbf b$ and $\mathbf b$ are linearly dependent since
$\mathbf a=\mathbf b+\mathbf c\,$. Thus:
$\mathbf a-\mathbf b-\mathbf c=\mnul\,$.
Exercise 8: Change of Base and Coordinates in the Plane
In this exercise we consider how the coordinates of a given vector changes when we change the basis.
The figure shows an ordinary basis $e=(\mathbf i, \mathbf j)$ and a basis $a=(\mathbf a_1, \mathbf a_2)$.
V
A vector $\mathbf u$ has the coordinates $(5,-1)$ with respect to the basis e. Determine the coordinates of $\mathbf u$ with repsect to basis a.
A vector $\mathbf v$ has the coordinates $(-1,-2)$ with respect to basis a. Determine the coordinates of $\mathbf v$ with repsect to basis e.
answer
$_\mathrm a\mathbf u=
\begin{matr}{r}2\\3\end{matr}$ and $ _\mathrm e \mathbf v=
\begin{matr}{r}-3\\0\end{matr}.$
Exercise 9: Change of Base and Coordinates in Space
In this exercise we work both with an ordinary coordinate system and with the basis $a$ that is shown in the figure.
W
Determine the determinant of the matrix $\,\left[\,\ma_1\,\,\ma_2\,\,\ma_3\,\right]\,.$ Explain that the set $\,(\ma_1,\ma_2,\ma_3)\,$ actually constitutes a basis.
hint
Read the coordinates for the three vectors with respect to the ordinary basis.
answer
Since there are three vectors, and since they are linearly independent, they constitute a basis for the set of vectors in space.
X
Three space vectors $\mathbf u,\,\mv$ and $\mathbf w$ are known from their coordinates with respect to basis a like this:
Determine an equation for $\alpha$ with respect to the given ordinary coordinate system.
hint
You get a normal vector for $\alpha$ by making the cross product of the two direction vectors.
answer
The cross product of the two direction vectors are $(-3,-1,7)$. An initial point in the plane is chosen: $(0,-2,0)$. Thus the equation for the plane is:
