Proof

Suppose that a matrix A is $n \times n$ matrix. Then if a matrix is multiplied by k, the determinant is multiplied by $k^{n}$ $| k (\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}) | = k^{3} | \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} |$

We prove when $n = 3$ .

Since the property of (2), we have

$| \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ k a_{31} & k a_{32} & k a_{33} \end{array} | = k | \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} |$

Therefore, we obtain

$\begin{array}{rcl} (LHS) & = & | k (\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}) | \\ = & | \begin{array}{ccc} k a_{11} & k a_{12} & k a_{13} \\ k a_{21} & k a_{22} & k a_{23} \\ k a_{31} & k a_{32} & k a_{33} \end{array} | \\ = & k | \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ k a_{21} & k a_{22} & k a_{23} \\ k a_{31} & k a_{32} & k a_{33} \end{array} | ( ∵ Property of Determinant (2) ) \\ = & k^{2} | \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ k a_{31} & k a_{32} & k a_{33} \end{array} | ( ∵ Property of Determinant (2) ) \\ = & k^{3} | \begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array} | ( ∵ Property of Determinant (2) ) \\ = & (RHS) \end{array}$

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Proof