studybyyourself

Question 1

Which element of $A$ is equal to -4 ?
$A=\begin{pmatrix} 1 & -3 & -1 \\ -3 & 0 & -8 \\ 7 & -4 & -5 \\ \end{pmatrix}$

$a_{2,3}$

$a_{1,3}$

$a_{2,1}$

$a_{3,2}$

Question 2

Which matricial equation is equivalent to this system ?
$\left\{ \begin{array}{ r@{~}c@{~} r!{~} } -9x+3y & = & 3 \\ 6x+6y & = & -3 \\ \end{array} \right.$

$\begin{pmatrix} -9+3 \\ 6+6 \\ \end{pmatrix}= \begin{pmatrix} 3 \\ -3 \\ \end{pmatrix} $

$\begin{pmatrix} -9 & 3 \\ 6 & 6 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix}= \begin{pmatrix} 3 \\ -3 \\ \end{pmatrix}$

$\begin{pmatrix} -9 & 6 \\ 3 & 6 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix}= \begin{pmatrix} 3 \\ -3 \\ \end{pmatrix}$

$\begin{pmatrix} -9x & 6y \\ 3x & 6y \\ \end{pmatrix} = \begin{pmatrix} 3 \\ -3 \\ \end{pmatrix} $

Question 3

Which augmented matrix is in echelon form ?

$\left( \begin{array}{ c c c | c } -1 & 0 & -2 & 2 \\ 1 & 0 & 2 & -1 \\ 3 & 0 & 6 & 1 \end{array} \right)$

$\left( \begin{array}{ c c c | c } 1 & 0 & 2 & 0 \\ 0 & 1 & -3 & -1 \\ 0 & 0 & 0 & 0 \end{array} \right)$

$\left( \begin{array}{ c c c | c } -2 & 1 & 0 & 1 \\ 0 & -3 & 1 & 0 \\ 0 & 4 & 1 & 0 \end{array} \right)$

$\left( \begin{array}{ c c c | c } 0 & 1 & -3 & 0 \\ 0 & -1 & 0 & 0 \\ -3 & 1 & 1 & 0 \end{array} \right)$

Question 4

What are the solutions of this augmented matrix ?
$\left( \begin{array}{!{}l@{~} !{}l@{~} | r!{~} } 1 & 2 & -1 \\ 2 & 2 & 2 \\ \end{array} \right)$

$\{\}$

infinitely many solutions

$\{3,-2\}$

$\{-3,1\}$

Question 5

For which value of $a$ has this equation infinitely many solutions ?

$\begin{pmatrix} 1 & 2 \\ 2 & a \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix}$

$\{4\}$

infinitely many values

$\{-1\}$

$\{2\}$

Question 6

What are the solutions of this equation ?

$\begin{pmatrix} -18 & -6 \\ 6 & 2 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} -6 \\ 1 \\ \end{pmatrix}$

$\{\}$

infinitely many solutions

$\{ (\frac{1}{8}, \frac{3}{8}) \}$

$\{(0,1)\}$

Question 7

Which equation is $\vec{x}$ the solution thereof ?
$\vec{x}= \begin{pmatrix} -2 \beta \\ -1+3\beta \\ \beta \\ \end{pmatrix}, \beta \in \mathbb{R}$

$\left( \begin{smallmatrix} 1 & 0 & 2 \\ 2 & 1 & 1 \\ 3 & 0 & 6 \end{smallmatrix} \right) \left( \begin{smallmatrix} x \\ y \\ z \end{smallmatrix} \right) = \left( \begin{smallmatrix} 0 \\ -1 \\ 0 \end{smallmatrix} \right)$

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

$\left( \begin{smallmatrix} -1 & 0 & -2 \\ 1 & 0 & 2 \\ 3 & 0 & 6 \end{smallmatrix} \right) \left( \begin{smallmatrix} x \\ y \\ z \end{smallmatrix} \right) = \left( \begin{smallmatrix} 2 \\ -1 \\ 1 \end{smallmatrix} \right)$

$\left( \begin{smallmatrix} -2 & 1 & 0 \\ 0 & -3 & 1 \\ 0 & 4 & 1 \end{smallmatrix} \right) \left( \begin{smallmatrix} x \\ y \\ z \end{smallmatrix} \right) = \left( \begin{smallmatrix} 1 \\ 0 \\ 0 \end{smallmatrix} \right)$

3.1