diff --git a/docs/en/mathematics/linear-algebra/determinants.md b/docs/en/mathematics/linear-algebra/determinants.md index b192f78..ddfb1b8 100644 --- a/docs/en/mathematics/linear-algebra/determinants.md +++ b/docs/en/mathematics/linear-algebra/determinants.md @@ -124,22 +124,22 @@ $$ ??? note "*Proof*:" -If $i = j$ then we obtain the cofactor expansion of $\det(A)$ along the $i$th row of $A$. + If $i = j$ then we obtain the cofactor expansion of $\det(A)$ along the $i$th row of $A$. -If $i \neq j$, let $A^*$ be the matrix obtained by replacing the $j$th row of $A$ by the $i$th row of $A$ + If $i \neq j$, let $A^*$ be the matrix obtained by replacing the $j$th row of $A$ by the $i$th row of $A$ -$$ - A^* = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ \vdots \\ a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots \\ a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix} \begin{array}{ll} j\text{th row}\\ \\ \\ \\\end{array} -$$ + $$ + A^* = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ \vdots \\ a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots \\ a_{11} & a_{12} & \cdots & a_{1n} \\ \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix} \begin{array}{ll} j\text{th row}\\ \\ \\ \\\end{array} + $$ -since two rows of $A^*$ are the same its determinant must be zero. It follows from the cofactor expansion of $\det(A^*)$ along the $j$th row that + since two rows of $A^*$ are the same its determinant must be zero. It follows from the cofactor expansion of $\det(A^*)$ along the $j$th row that -$$ -\begin{align*} - 0 &= \det(A^*) = a_{i1} A_{j1}^* + a_{i2} A_{j2}^* + \dots + a_{in} A_{jn}^*, \\ - &= a_{i1} A_{j1} + a_{i2} A_{j2} + \dots + a_{in} A_{jn}. -\end{align*} -$$ + $$ + \begin{align*} + 0 &= \det(A^*) = a_{i1} A_{j1}^* + a_{i2} A_{j2}^* + \dots + a_{in} A_{jn}^*, \\ + &= a_{i1} A_{j1} + a_{i2} A_{j2} + \dots + a_{in} A_{jn}. + \end{align*} + $$ > *Theorem*: let $E$ be an $n \times n$ elementary matrix and $A$ an $n \times n$ matrix with $n \in \mathbb{N}$ then we have >