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Category: Matrices and Determinants

E-is-a-vectorial-plan-in-R-with-a-base-B-i-j-f-is-an-endomorphism-of-E-defined-u-xi-yj-by-f-u-7x-12y-i-4x-7y-j-1-Determinate-f-i-and-f-j-then-write-the-mat

Question Number 86021 by mathocean1 last updated on 26/Mar/20 $${E}\:{is}\:{a}\:{vectorial}\:{plan}\:{in}\:\mathbb{R}\:{with}\:{a}\:{base} \\ $$$${B}=\left(\overset{\rightarrow} {{i}},\overset{\rightarrow} {{j}}\right).\:{f}\:{is}\:{an}\:{endomorphism}\:{of}\:{E} \\ $$$${defined}\:\forall\:\overset{\rightarrow} {{u}}={x}\overset{\rightarrow} {{i}}+{y}\overset{\rightarrow} {{j}}\:{by}\:{f}\left(\overset{\rightarrow} {{u}}\right)=\left(−\mathrm{7}{x}−\mathrm{12}{y}\right)\overset{\rightarrow} {{i}}+\left(\mathrm{4}{x}+\mathrm{7}{y}\right)\overset{\rightarrow} {{j}}. \\ $$$$\left.\mathrm{1}\right)\:{Determinate}\:{f}\left(\overset{\rightarrow} {{i}}\right)\:{and}\:{f}\left(\overset{\rightarrow}…

Question-151028

Question Number 151028 by kalenis last updated on 17/Aug/21 Answered by amin96 last updated on 17/Aug/21 $$\begin{cases}{{x}+{y}−{z}=\mathrm{7}\:\:\:\:\:−}\\{{x}−{y}+\mathrm{2}{z}=\mathrm{3}}\\{\mathrm{2}{x}+{y}+{z}=\mathrm{9}}\end{cases}\Rightarrow\begin{cases}{\mathrm{2}{x}+{z}=\mathrm{10}}\\{\mathrm{10}+{y}=\mathrm{9}}\end{cases}\:\Rightarrow\begin{cases}{\mathrm{2}{x}+{z}=\mathrm{10}}\\{{y}=−\mathrm{1}}\end{cases} \\ $$$$\begin{cases}{{x}−{z}=\mathrm{8}}\\{{x}+\mathrm{2}{z}=\mathrm{2}}\end{cases}\:\:\begin{cases}{\mathrm{2}{x}−\mathrm{2}{z}=\mathrm{16}}\\{{x}+\mathrm{2}{z}=\mathrm{2}}\end{cases}\Rightarrow\mathrm{3}{x}=\mathrm{18}\:\:{x}=\mathrm{6} \\ $$$${x}−{z}=\mathrm{8}\:\:\Rightarrow\:\:{z}=−\mathrm{2}\:\:\:\:\:{answer}\:\:\left(\mathrm{6};\:\:−\mathrm{1};\:\:−\mathrm{2}\right) \\ $$ Terms of…

Let-A-and-B-is-3-3-matrix-of-equal-number-where-A-symmetric-matrix-B-skew-symmetric-matrix-and-the-relation-A-B-A-B-A-B-A-B-then-the-value-of-k-AB-T-1-k-AB-a-1-

Question Number 19589 by gourav~ last updated on 13/Aug/17 $${Let}\:{A}\:{and}\:{B}\:{is}\:\mathrm{3}×\mathrm{3}\:{matrix}\:{of}\:{equal}\:{number} \\ $$$${where}\:{A}={symmetric}\:{matrix}\: \\ $$$$….{B}={skew}\:{symmetric}\:{matrix} \\ $$$${and}\:{the}\:{relation}…\:\left({A}+{B}\right)\left({A}−{B}\right)=\left({A}−{B}\right)\left({A}+{B}\right) \\ $$$${then}..{the}\:{value}\:{of}..\:…\:{k} \\ $$$$\:\:\:\:\left({AB}\right)^{{T}} =\left(−\mathrm{1}\right)^{{k}} \left({AB}\right) \\ $$$$\left({a}\right)\:−\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({c}\right)\:\mathrm{2} \\…

A-matrix-has-N-rows-and-2k-1-columns-Each-column-is-filled-with-M-ones-and-N-M-zeros-A-given-row-j-is-cool-if-and-only-if-i-1-2k-1-a-ji-k-Find-the-minimum-and-the-maximum-number-of-c

Question Number 19547 by dioph last updated on 12/Aug/17 $$\mathrm{A}\:\mathrm{matrix}\:\mathrm{has}\:{N}\:\mathrm{rows}\:\mathrm{and}\:\mathrm{2}{k}−\mathrm{1}\: \\ $$$$\mathrm{columns}.\:\mathrm{Each}\:\mathrm{column}\:\mathrm{is}\:\mathrm{filled}\:\mathrm{with} \\ $$$${M}\:\mathrm{ones}\:\mathrm{and}\:{N}−{M}\:\mathrm{zeros}. \\ $$$$\mathrm{A}\:\mathrm{given}\:\mathrm{row}\:{j}\:\mathrm{is}\:“{cool}''\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if} \\ $$$$\underset{{i}=\mathrm{1}} {\overset{\mathrm{2}{k}−\mathrm{1}} {\sum}}{a}_{{ji}} \:\geqslant\:{k}.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{and} \\ $$$$\mathrm{the}\:\mathrm{maximum}\:\mathrm{number}\:\mathrm{of}\:\mathrm{cool}\:\mathrm{rows} \\ $$$$\mathrm{for}\:\mathrm{given}\:{N},\:{k}\:\mathrm{and}\:{M}.…

If-A-is-a-3-3-matrix-where-det-A-2-then-what-will-be-det-3A-2-A-1-knowing-that-A-1-is-the-inverse-of-A-

Question Number 149116 by bramlexs22 last updated on 03/Aug/21 $$\:\mathrm{If}\:\mathrm{A}\:\mathrm{is}\:\mathrm{a}\:\mathrm{3}×\mathrm{3}\:\mathrm{matrix}\:\mathrm{where}\:\mathrm{det}\left(\mathrm{A}\right)=−\mathrm{2} \\ $$$$\mathrm{then}\:\mathrm{what}\:\mathrm{will}\:\mathrm{be}\:\mathrm{det}\left(\mathrm{3A}^{\mathrm{2}} \mathrm{A}^{−\mathrm{1}} \right)? \\ $$$$\mathrm{knowing}\:\mathrm{that}\:\mathrm{A}^{−\mathrm{1}} \:\mathrm{is}\:\mathrm{the}\:\mathrm{inverse} \\ $$$$\mathrm{of}\:\mathrm{A}\: \\ $$ Answered by EDWIN88 last…