Question Number 143718 by cherokeesay last updated on 17/Jun/21

Commented by TheHoneyCat last updated on 17/Jun/21

$${Hi},\:{what}\:{do}\:{you}\:{mean}\:{by}\:''\mid{B}\mid''? \\ $$$${it}\:{looks}\:{like}\:{you}\:{are}\:{talking}\:{about}\:{a}\:{norm}\:{but}\:{which}\:{one}? \\ $$
Commented by TheHoneyCat last updated on 17/Jun/21
![Q_2 ED= [(1,X),(X,1) ]× [(2,y),(1,(−y)) ]= [((2+X),((1−X)y)),((2X+1),((X−1)y)) ] DE= [(2,y),(1,(−y)) ]× [(1,X),(X,1) ]= [((2+yX),(2X+y)),((1−yX),(X−y)) ] the first term (m_(0,0) ) imposes y=1 to get DE=ED the one under (m_(1,0) ) imposes 2X+1=1−X⇔3X=0⇔X=0 checking (m_(0,1) ) with these conditions, we get 1=−1 which is obviously false so ∀(X,y)∈C^2 ED≠DE](https://www.tinkutara.com/question/Q143723.png)
$${Q}_{\mathrm{2}} \\ $$$${ED}=\begin{bmatrix}{\mathrm{1}}&{{X}}\\{{X}}&{\mathrm{1}}\end{bmatrix}×\begin{bmatrix}{\mathrm{2}}&{{y}}\\{\mathrm{1}}&{−{y}}\end{bmatrix}=\begin{bmatrix}{\mathrm{2}+{X}}&{\left(\mathrm{1}−{X}\right){y}}\\{\mathrm{2}{X}+\mathrm{1}}&{\left({X}−\mathrm{1}\right){y}}\end{bmatrix} \\ $$$${DE}=\begin{bmatrix}{\mathrm{2}}&{{y}}\\{\mathrm{1}}&{−{y}}\end{bmatrix}×\begin{bmatrix}{\mathrm{1}}&{{X}}\\{{X}}&{\mathrm{1}}\end{bmatrix}=\begin{bmatrix}{\mathrm{2}+{yX}}&{\mathrm{2}{X}+{y}}\\{\mathrm{1}−{yX}}&{{X}−{y}}\end{bmatrix} \\ $$$$\mathrm{the}\:\mathrm{first}\:\mathrm{term}\:\left({m}_{\mathrm{0},\mathrm{0}} \right)\:\mathrm{imposes}\:{y}=\mathrm{1}\:\mathrm{to}\:\mathrm{get}\:\mathrm{D}{E}={ED} \\ $$$$\mathrm{the}\:\mathrm{one}\:\mathrm{under}\:\left({m}_{\mathrm{1},\mathrm{0}} \right)\:\mathrm{imposes}\:\mathrm{2}{X}+\mathrm{1}=\mathrm{1}−{X}\Leftrightarrow\mathrm{3}{X}=\mathrm{0}\Leftrightarrow{X}=\mathrm{0} \\ $$$$\mathrm{checking}\:\left({m}_{\mathrm{0},\mathrm{1}} \right)\:\mathrm{with}\:\mathrm{these}\:\mathrm{conditions},\:\mathrm{we}\:\mathrm{get}\:\mathrm{1}=−\mathrm{1} \\ $$$$\mathrm{which}\:\mathrm{is}\:\mathrm{obviously}\:\mathrm{false} \\ $$$$ \\ $$$$\mathrm{so}\:\forall\left({X},{y}\right)\in\mathbb{C}^{\mathrm{2}} \:{ED}\neq{DE} \\ $$
Commented by Olaf_Thorendsen last updated on 17/Jun/21

$$\mathrm{In}\:\mathrm{some}\:\mathrm{countries},\:\mathrm{France}\:\mathrm{for}\:\mathrm{example}, \\ $$$$\mathrm{det}\left(\mathrm{B}\right)\:\mathrm{is}\:\mathrm{sometimes}\:\mathrm{noted}\:\mid\mathrm{B}\mid. \\ $$$$ \\ $$
Commented by TheHoneyCat last updated on 17/Jun/21

$$\mathrm{well},\:\mathrm{I}\:\mathrm{think}\:\mathrm{this}\:\mathrm{notation}\:\mathrm{only}\:\mathrm{concers}\:\mathrm{the}\:\mathrm{expressions} \\ $$$$\mathrm{it}\:\mathrm{is}\:\mathrm{used}\:\mathrm{to}\:\mathrm{do}\:\mathrm{explicit}\:\mathrm{calculations}\:\mathrm{or}\:\mathrm{block}\:\mathrm{calculations} \\ $$$$ \\ $$$$\mathrm{At}\:\mathrm{least}\:\mathrm{that}'\mathrm{s}\:\mathrm{how}\:\mathrm{we}\:\mathrm{do}\:\mathrm{it}\:\mathrm{in}\:\mathrm{my}\:\mathrm{School}\:\left(\mathrm{which}\:\mathrm{is}\:\mathrm{in}\:\mathrm{France}\:\mathrm{of}\:\mathrm{course},\:\mathrm{otherwise}\:\mathrm{I}\:\mathrm{would}\:\mathrm{not}\:\mathrm{have}\:\mathrm{any}\:\mathrm{idea}\right) \\ $$$$\mathrm{we}\:\mathrm{use}\:\mathrm{det}{M}\:\mathrm{when}\:{M}\:\mathrm{is}\:\mathrm{not}\:\mathrm{explicit} \\ $$$$ \\ $$$$\mathrm{But}\:\mathrm{maybe}\:\mathrm{is}\:\mathrm{is}\:\mathrm{used}\:\mathrm{is}\:\mathrm{some}\:\mathrm{other}\:\mathrm{places} \\ $$
Commented by TheHoneyCat last updated on 17/Jun/21

$$\mathrm{det}\left({B}\right)=\left(−\mathrm{2}{i}\right)\left(\mathrm{6}+\mathrm{2}{i}\right)−\mathrm{4}×\mathrm{1} \\ $$$$=−\mathrm{12}{i}+\mathrm{4}−\mathrm{4} \\ $$$$=−\mathrm{12}{i} \\ $$