Linearità

$$ \begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{vmatrix} = \begin{vmatrix} \sum_{j=1}^n k_j a_{lj} & \sum_{j=1}^n k_j a_{lj} & \ldots & \sum_{j=1}^n k_j a_{lj}\\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{vmatrix} $$ $$ \begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{vmatrix} = \begin{vmatrix} \sum_{j=1}^n k_j a_{lj} & \sum_{j=1}^n k_j a_{lj} & \ldots & \sum_{j=1}^n k_j a_{lj}\\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{vmatrix} $$ $$ \begin{vmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{vmatrix} $$ $$ \downarrow $$ $$ \small \begin{vmatrix} \sum_{j=1}^n k_j a_{lj} & \sum_{j=1}^n k_j a_{lj} & \ldots & \sum_{j=1}^n k_j a_{lj}\\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{vmatrix} $$

Se sostituiamo al posto di una qualunque riga o colonna, una combinazione lineare di altre righe o colonne con opportuni coefficienti, il determinante rimane invariato

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