studybyyourself

Question 1

What is the comatrix of $A$ ?
$\begin{equation*} A = \begin{pmatrix} 0 & -1 & 1 \\ -4 & 0 & -5 \\ -7 & 3 & 0 \end{pmatrix} \end{equation*}$

$\left( \begin{smallmatrix} 15 & 35 & -12 \\ 3 & 7 & -7 \\ 5 & -4 & -4 \end{smallmatrix} \right)$

$\left( \begin{smallmatrix} 15 & 35 & -12 \\ 3 & 7 & 7 \\ 5 & -4 & -4 \end{smallmatrix} \right)$

$\left( \begin{smallmatrix} 15 & 35 & -12 \\ 3 & 7 & 7 \\ -5 & -4 & -4 \end{smallmatrix} \right)$

$\left( \begin{smallmatrix} 15 & 35 & 12 \\ 3 & 7 & 7 \\ 5 & -4 & -4 \end{smallmatrix} \right)$

Question 2

Let $B^{T}= \begin{pmatrix} 3 & -4 & -11 \\ 13 & 9 & -40 \end{pmatrix} $. What is $B$ equal to ?

$\left( \begin{smallmatrix} 3 \\ -4 \\ -11 \end{smallmatrix}\right)$

$\left( \begin{smallmatrix} 13 \\ 9 \\ -40 \end{smallmatrix}\right)$

$\left( \begin{smallmatrix} 13 & 3 \\ 9 & -4 \\ -40 & -11 \end{smallmatrix}\right)$

$\left( \begin{smallmatrix} 3 & 13 \\ -4 & 9 \\ -11 & -40 \end{smallmatrix}\right)$

Question 3

What matrix is neutral for matrix multiplication ?

null matrix

matrix inverse

identity matrix

matrix of cofactors

Question 4

Which function is bijective ?

$a \colon \, \mathbb{R} \to \mathbb{R}$, $ x \mapsto \log(x)$

$b \colon \, \mathbb{R} \to \mathbb{R}$, $ x \mapsto sin(x)$

$c \colon \, \mathbb{R} \to \mathbb{R}$, $ x \mapsto x^2$

$d \colon \, \mathbb{R} \to \mathbb{R}$, $ x \mapsto x$

Question 5

What is $A$ equal to ?

$\begin{equation*} I = A \begin{pmatrix} 0 & -1 & 1 \\ 1 & 0 & 0 \\ -1 & 1 & 0 \end{pmatrix} \end{equation*}$

$\left( \begin{smallmatrix} 0 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{smallmatrix} \right)$

$\left( \begin{smallmatrix} 0 & 1 & 0 \\ 1 & 1 & -1 \\ 3 & 3 & 3 \end{smallmatrix} \right)$

$\left( \begin{smallmatrix} 1 & 1 & -3 \\ 3 & 3 & -3 \\ 1 & 1 & 1 \end{smallmatrix} \right)$

$\left( \begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{smallmatrix} \right)$

Question 6

Let $t$ be a linear transformation. What is the pre-image of $\vec{y}=\left( \begin{smallmatrix} -1 \\ -4 \end{smallmatrix} \right)$ under $t$ ?

$\begin{equation*} t \colon \, \mathbb{R}^2 \to \mathbb{R}^2 \\ \quad \vec{x} \mapsto \begin{pmatrix} 3 & -1 \\ -1 & 0 \end{pmatrix} \vec{x} \end{equation*}$

$\begin{pmatrix} -2 \\ 4 \end{pmatrix}$

$\begin{pmatrix} 4 \\ 13 \end{pmatrix}$

$\begin{pmatrix} -1 \\ 0 \end{pmatrix}$

$\{\}$

Question 7

Which linear transformation is bijective ?

$e \colon \, \mathbb{R}^3 \to \mathbb{R}^3$, $ \vec{x} \mapsto \left( \begin{smallmatrix} 1 & 3 & -1 \\ 0 & 0 & 1 \\ -2 & -4 & 1 \end{smallmatrix} \right)\vec{x}$

$f \colon \, \mathbb{R}^2 \to \mathbb{R}^2$, $ \vec{x} \mapsto \left( \begin{smallmatrix} 0 & 0 \\ 0 & -9 \end{smallmatrix} \right)\vec{x}$

$g \colon \, \mathbb{R}^3 \to \mathbb{R}$, $ \vec{x} \mapsto \left( \begin{smallmatrix} 1 & 3 & -1 \end{smallmatrix} \right)\vec{x}$

$h \colon \, \mathbb{R}^3 \to \mathbb{R}^2$, $ \vec{x} \mapsto \left( \begin{smallmatrix} 1 & 3 & -1 \\ 0 & 0 & 1 \end{smallmatrix} \right)\vec{x}$

10.1