Question

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

Answer 1

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

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

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

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

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

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

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

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

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

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

$=2$

Answer 2

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