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Category: Algebra

Question-204211

Question Number 204211 by mnjuly1970 last updated on 08/Feb/24 Answered by witcher3 last updated on 08/Feb/24 $$\mathrm{detA}=\mathrm{7}=\overset{−} {\mathrm{0}}\:\:\mathrm{in}\:\mathbb{Z}/\mathrm{7}\mathbb{Z} \\ $$$$\mathrm{A}\notin\mathrm{GL}\left(\mathrm{2};\mathbb{Z}/\mathrm{7}\mathbb{Z}\right) \\ $$$$ \\ $$ Commented…

Question-204179

Question Number 204179 by mr W last updated on 08/Feb/24 Answered by AST last updated on 08/Feb/24 $${f}\left({x}\right)={f}\left({y}\right)\Rightarrow{f}\left({f}\left({x}\right)\right)={f}\left({f}\left({y}\right)\right)\Rightarrow{x}^{\mathrm{2}} −{x}={y}^{\mathrm{2}} −{y} \\ $$$$\Rightarrow\left({x}−{y}\right)\left({x}+{y}\right)={x}−{y}\Rightarrow{x}=\mathrm{1}−{y}\:{or}\:{x}={y} \\ $$$$\Rightarrow{f}\left({x}\right)={f}\left(\mathrm{1}−{x}\right)\Rightarrow{f}\left(\mathrm{0}\right)={f}\left(\mathrm{1}\right) \\…

Question-204158

Question Number 204158 by siyathokoza last updated on 07/Feb/24 Commented by siyathokoza last updated on 07/Feb/24 $$\boldsymbol{\mathrm{HI}}\:\boldsymbol{\mathrm{PROF}}… \\ $$$$\boldsymbol{\mathrm{Please}}\:\boldsymbol{\mathrm{explain}}\:\boldsymbol{\mathrm{this}}\:\boldsymbol{\mathrm{word}}\:\boldsymbol{\mathrm{problem}} \\ $$ Commented by siyathokoza last…

the-maximum-value-of-f-x-y-xy-x-3-y-2-attained-over-the-square-0-x-1-0-y-1-is-

Question Number 204152 by universe last updated on 07/Feb/24 $$\:\:\mathrm{the}\:\mathrm{maximum}\:\mathrm{value}\:\mathrm{of}\:\mathrm{f}\left(\mathrm{x},\mathrm{y}\right)\:=\:\mathrm{xy}−\mathrm{x}^{\mathrm{3}} \mathrm{y}^{\mathrm{2}} \\ $$$$\:\mathrm{attained}\:\mathrm{over}\:\mathrm{the}\:\mathrm{square}\:\mathrm{0}\leqslant\mathrm{x}\leqslant\mathrm{1};\mathrm{0}\leqslant\mathrm{y}\leqslant\mathrm{1}\:\mathrm{is} \\ $$ Answered by ajfour last updated on 08/Feb/24 $${f}\left({x},{y}\right)={z}={t}−{t}^{\mathrm{3}} \\ $$$$\frac{{dz}}{{dt}}=\mathrm{1}−\mathrm{3}{t}^{\mathrm{2}}…

a-b-x-y-R-a-b-23-ax-by-79-ax-2-by-2-217-ax-3-by-3-661-Find-ax-4-by-4-

Question Number 204129 by hardmath last updated on 06/Feb/24 $$\mathrm{a}\:,\:\mathrm{b}\:,\:\mathrm{x}\:,\:\mathrm{y}\:\in\:\mathbb{R} \\ $$$$\mathrm{a}\:+\:\mathrm{b}\:=\:\mathrm{23} \\ $$$$\mathrm{ax}\:+\:\mathrm{by}\:=\:\mathrm{79} \\ $$$$\mathrm{ax}^{\mathrm{2}} \:+\:\mathrm{by}^{\mathrm{2}} \:=\:\mathrm{217} \\ $$$$\mathrm{ax}^{\mathrm{3}} \:+\:\mathrm{by}^{\mathrm{3}} \:=\:\mathrm{661} \\ $$$$\mathrm{Find}:\:\:\:\mathrm{ax}^{\mathrm{4}} \:+\:\mathrm{by}^{\mathrm{4}}…

x-2y-z-8-3x-2y-z-10-4x-3y-2z-4-Solve-with-the-help-of-matrix-

Question Number 204056 by hardmath last updated on 05/Feb/24 $$\begin{cases}{\mathrm{x}\:+\:\mathrm{2y}\:+\:\mathrm{z}\:=\:\mathrm{8}}\\{\mathrm{3x}\:+\:\mathrm{2y}\:+\:\mathrm{z}\:=\:\mathrm{10}}\\{\mathrm{4x}\:+\:\mathrm{3y}\:−\:\mathrm{2z}\:=\:\mathrm{4}}\end{cases} \\ $$$$\mathrm{Solve}\:\mathrm{with}\:\mathrm{the}\:\mathrm{help}\:\mathrm{of}\:\mathrm{matrix} \\ $$ Answered by AST last updated on 05/Feb/24 $$\begin{bmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{1}}\\{\mathrm{3}}&{\mathrm{2}}&{\mathrm{1}}\\{\mathrm{4}}&{\mathrm{3}}&{−\mathrm{2}}\end{bmatrix}\begin{bmatrix}{{x}}\\{{y}}\\{{z}}\end{bmatrix}=\begin{bmatrix}{\mathrm{8}}\\{\mathrm{10}}\\{\mathrm{4}}\end{bmatrix}\underset{{R}_{\mathrm{3}} −\mathrm{4}{R}_{\mathrm{1}} } {\overset{{R}_{\mathrm{2}}…

y-sinx-cos-3-x-find-y-

Question Number 204054 by hardmath last updated on 05/Feb/24 $$\mathrm{y}\:=\:\sqrt{\mathrm{sinx}}\:+\:\mathrm{cos}^{\mathrm{3}} \mathrm{x} \\ $$$$\mathrm{find}:\:\:\mathrm{y}^{'} \:=\:? \\ $$ Answered by AST last updated on 05/Feb/24 $${y}'=\frac{{cos}\left({x}\right)}{\:\mathrm{2}\sqrt{{sinx}}}−\mathrm{3}{sin}\left({x}\right){cos}^{\mathrm{2}} {x}…