Determinant/Zero, linear dependent and rank property/Fact/Proof

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Proof

The relation between rank, invertibility and linear independence was proven in

Suppose now that the rows are linearly dependent. After exchanging rows, we may assume that Failed to parse (syntax error): {\displaystyle {{}} v_n = \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Determinant/Zero, linear dependent and rank property/Fact/Proof]] __NOINDEX__ 1}^{n-1} s_i v_i} . Then, due to

and

we get

Failed to parse (unknown function "\begin{pmatrix}"): {\displaystyle {{}} \det M = \det \begin{pmatrix} v_1 \\ \vdots \\ v_{n-1} \\ \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Determinant/Zero, linear dependent and rank property/Fact/Proof]] __NOINDEX__ 1}^{n-1} s_i v_i \end{pmatrix} = \sum_{i <table class="metadata plainlinks ambox ambox-notice" style=""> <tr> <td class="mbox-image"><div style="width: 52px;"> [[File:Wikiversity logo 2017.svg|50px|link=]]</div></td> <td class="mbox-text" style=""> '''[[m:Soft redirect|Soft redirect]]'''<br />This page can be found at <span id="SoftRedirect">[[mw:Help:Magic words#Other]]</span>. </td> </tr> </table>[[Category:Wikiversity soft redirects|Determinant/Zero, linear dependent and rank property/Fact/Proof]] __NOINDEX__ 1}^{n-1} s_i \det \begin{pmatrix} v_1 \\ \vdots \\ v_{n-1} \\ v_i \end{pmatrix} = 0 \, . }

Now suppose that the rows are linearly independent. Then, by exchanging of rows, scaling and addition of a row to another row, we can transform the matrix successively into the identity matrix. During these manipulations, the determinant is multiplied with some factor . Since the determinant of the identity matrix is , the determinant of the initial matrix is .