Science, asked by thapaavinitika6765, 8 months ago

Solve:\mathrm{Rank}\:\begin{pmatrix}2&1&6\\ 3&4&5\end{pmatrix}=2

Answers

Answered by Anonymous
2

\mathrm{Rank}\:\begin{pmatrix}2&1&6\\ 3&4&5\end{pmatrix}=2

\mathrm{Reduce\:matrix\:to\:reduced\:row\:echelon\:form}\:\begin{pmatrix}1\:&\:\cdots \:&\:b\:\\ 0\:&\ddots \:&\:\vdots \\ 0\:&\:0\:&\:1\end{pmatrix}

\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_2\:\mathrm{\:by\:performing}\:R_2\:\leftarrow \:R_2-\frac{2}{3}\cdot \:R_1

=\begin{pmatrix}3&4&5\\ 0&-\frac{5}{3}&\frac{8}{3}\end{pmatrix}

\mathrm{Reduce\:matrix\:to\:reduced\:row\:echelon\:form}\:\begin{pmatrix}1\:&\:\cdots \:&\:b\:\\ 0\:&\ddots \:&\:\vdots \\ 0\:&\:0\:&\:1\end{pmatrix}

\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_1\:\mathrm{\:by\:performing}\:R_1\:\leftarrow \:R_1-4\cdot \:R_2

=\begin{pmatrix}3&0&\frac{57}{5}\\ 0&1&-\frac{8}{5}\end{pmatrix}

\mathrm{Multiply\:matrix\:row\:by\:constant:}\:R_1\:\leftarrow \frac{1}{3}\cdot \:R_1

=\begin{pmatrix}1&0&\frac{19}{5}\\ 0&1&-\frac{8}{5}\end{pmatrix}

\bf{The\:rank\:of\:a\:matrix\:is\:the\:number\:of\:non\:all-zeros\:rows}

=2

Answered by XxMrGlamorousXx
0

\begin{gathered}\mathrm{Rank}\:\begin{pmatrix}2&amp;1&amp;6\\ 3&amp;4&amp;5\end{pmatrix}=2\end{gathered}Rank(231465)=2</p><p></p><p>\begin{gathered}\mathrm{Reduce\:matrix\:to\:reduced\:row\:echelon\:form}\:\begin{pmatrix}1\:&amp;\:\cdots \:&amp;\:b\:\\ 0\:&amp;\ddots \:&amp;\:\vdots \\ 0\:&amp;\:0\:&amp;\:1\end{pmatrix}\end{gathered}Reducematrixtoreducedrowechelonform⎝⎜⎜⎛100⋯⋱0b⋮1⎠⎟⎟⎞</p><p></p><p>\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_2\:\mathrm{\:by\:performing}\:R_2\:\leftarrow \:R_2-\frac{2}{3}\cdot \:R_1CancelleadingcoefficientinrowR2byperformingR2←R2−32⋅R1</p><p></p><p>\begin{gathered}=\begin{pmatrix}3&amp;4&amp;5\\ 0&amp;-\frac{5}{3}&amp;\frac{8}{3}\end{pmatrix}\end{gathered}=(304−35538)</p><p></p><p>\begin{gathered}\mathrm{Reduce\:matrix\:to\:reduced\:row\:echelon\:form}\:\begin{pmatrix}1\:&amp;\:\cdots \:&amp;\:b\:\\ 0\:&amp;\ddots \:&amp;\:\vdots \\ 0\:&amp;\:0\:&amp;\:1\end{pmatrix}\end{gathered}Reducematrixtoreducedrowechelonform⎝⎜⎜⎛100⋯⋱0b⋮1⎠⎟⎟⎞</p><p></p><p>\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_1\:\mathrm{\:by\:performing}\:R_1\:\leftarrow \:R_1-4\cdot \:R_2CancelleadingcoefficientinrowR1byperformingR1←R1−4⋅R2</p><p></p><p>\begin{gathered}=\begin{pmatrix}3&amp;0&amp;\frac{57}{5}\\ 0&amp;1&amp;-\frac{8}{5}\end{pmatrix}\end{gathered}=(3001557−58)</p><p></p><p>\mathrm{Multiply\:matrix\:row\:by\:constant:}\:R_1\:\leftarrow \frac{1}{3}\cdot \:R_1Multiplymatrixrowbyconstant:R1←31⋅R1</p><p></p><p>\begin{gathered}=\begin{pmatrix}1&amp;0&amp;\frac{19}{5}\\ 0&amp;1&amp;-\frac{8}{5}\end{pmatrix}\end{gathered}=(1001519−58)</p><p></p><p>\bf{The\:rank\:of\:a\:matrix\:is\:the\:number\:of\:non\:all-zeros\:rows}Therankofamatrixisthenumberofnonall−zerosrows</p><p></p><p>=2=2</p><p></p><p>

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