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Answer:
dividing by a fraction is the same as multiplying by the reciprocal of the fraction.
In some cases of simplifying an algebraic expression, the expression will be a fraction. For example,
x2+3x
x+3
has a quadratic binomial in the numerator and a linear binomial in the denominator. We have to apply the different factorisation methods in order to factorise the numerator and the denominator before we can simplify the expression.
x2+3x
x+3
=
x(x+3)
x+3
=x(x≠−3)
If x=−3 then the denominator, x+3=0 and the fraction is undefined.
Explanation:
Hope it will help
\begin{gathered}\mathrm{Rank}\:\begin{pmatrix}2&1&6\\ 3&4&5\end{pmatrix}=2\end{gathered}
Rank(
2
3
1
4
6
5
)=2
\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
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\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_2\:\mathrm{\:by\:performing}\:R_2\:\leftarrow \:R_2-\frac{2}{3}\cdot \:R_1CancelleadingcoefficientinrowR
2
byperformingR
2
←R
2
−
3
2
⋅R
1
\begin{gathered}=\begin{pmatrix}3&4&5\\ 0&-\frac{5}{3}&\frac{8}{3}\end{pmatrix}\end{gathered}
=(
3
0
4
−
3
5
5
3
8
)
\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
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\mathrm{Cancel\:leading\:coefficient\:in\:row\:}\:R_1\:\mathrm{\:by\:performing}\:R_1\:\leftarrow \:R_1-4\cdot \:R_2CancelleadingcoefficientinrowR
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byperformingR
1
←R
1
−4⋅R
2
\begin{gathered}=\begin{pmatrix}3&0&\frac{57}{5}\\ 0&1&-\frac{8}{5}\end{pmatrix}\end{gathered}
=(
3
0
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57
−
5
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)
\mathrm{Multiply\:matrix\:row\:by\:constant:}\:R_1\:\leftarrow \frac{1}{3}\cdot \:R_1Multiplymatrixrowbyconstant:R
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←
3
1
⋅R
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\begin{gathered}=\begin{pmatrix}1&0&\frac{19}{5}\\ 0&1&-\frac{8}{5}\end{pmatrix}\end{gathered}
=(
1
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5
19
−
5
8
)
\bf{The\:rank\:of\:a\:matrix\:is\:the\:number\:of\:non\:all-zeros\:rows}Therankofamatrixisthenumberofnonall−zerosrows
=2=2