Proof by induction on $m$, the number of elementary row operations in the sequence $\hat o_1, \ldots, \hat o_m$. === Basis for the Induction === Suppose $m = 1$, so there is only one elementary row operation $\hat o$ in the sequence. Let $r_i$ denote the $i$'th row of $\mathbf A$. Suppose that $\hat o$ is of the type $r_i \to a r_i$, where $a \in R$ and $i \in \set {1, \ldots, n}$. From Effect of Elementary Row Operations on Determinant, it follows that: :$\map \det {\mathbf A} = a \map \det {\mathbf A'}$ Suppose that $\hat o$ is of the type $r_i \to r_i + ar_j$, where $a \in R$ and $i, j \in \set {1, \ldots, n}, i \ne j$. From Effect of Elementary Row Operations on Determinant, it follows that  :$\map \det {\mathbf A} = \map \det {\mathbf A'} = 1_R \map \det {\mathbf A'}$ where $1_R$ denotes the identity element of $\struct {R, \circ}$. Suppose that $\hat o$ is of the type $r_i \leftrightarrow r_j$. From Effect of Elementary Row Operations on Determinant, it follows that  :$\map \det {\mathbf A} = -\map \det {\mathbf A'} = -1_R \map \det {\mathbf A'}$ where the last equality follows from Product with Ring Negative: Corollary. This is the basis for the induction. === Induction Hypothesis === For $m \in \N$, let $\hat o_1, \ldots, \hat o_m$ be a finite sequence of elementary row operations.  This is the induction hypothesis: There exists $c \in R$ such that for all matrices of order $n$ $\mathbf A$: :$\map \det {\mathbf A} = c \map \det {\mathbf A'}$ where $\mathbf A'$ is the matrix of order $n$ that results from using the elementary row operations $\hat o_1, \ldots, \hat o_m$ on $\mathbf A$. === Induction Step === This is the induction step: Let $\hat o_1, \ldots, \hat o_m, \hat o_{m + 1}$ be a finite sequence of elementary row operations. Let $r_i$ denote the $i$'th row of $\mathbf A'$. Let $\mathbf A''$ denote the matrix of order $n$ that results from using the elementary row operation $\hat o_{m + 1}$ on $A'$. Then $\mathbf A''$ is equal to the matrix that results from using the elementary row operations $\hat o_1, \ldots, \hat o_m, \hat o_{m + 1}$ on $A$. Suppose that $\hat o_{m + 1}$ is of the type $r_i \to ar_i$, where $a \in R$ and $i \in \set {1, \ldots, n}$.  Then: {{begin-eqn}} {{eqn | l = \map \det {\mathbf A}       | r = c \map \det {\mathbf A'}       | c = by the induction hypothesis }} {{eqn | r = \paren {c a} \map \det {\mathbf A''}       | c = Effect of Elementary Row Operations on Determinant }} {{end-eqn}} Suppose that $\hat o_{m + 1}$ is of the type $r_i \to r_i + a r_j$, where $a \in R$ and $i, j \in \set {1, \ldots, n}, i \ne j$. Then: {{begin-eqn}} {{eqn | l = \map \det {\mathbf A}       | r = c \map \det {\mathbf A'}       | c = by the induction hypothesis }} {{eqn | r = c 1_R \map \det {\mathbf A''}       | c = Effect of Elementary Row Operations on Determinant }} {{eqn | r = c \map \det {\mathbf A''}       | c = {{Defof|Identity Element}}, since $R$ is commutative }} {{end-eqn}} Suppose that $\hat o_{m + 1}$ is of the type $r_i \leftrightarrow r_j$. Then: {{begin-eqn}} {{eqn | l = \map \det {\mathbf A}       | r = c \map \det {\mathbf A'}       | c = by the induction hypothesis }} {{eqn | r = c \paren {-1_R} \map \det {\mathbf A''}       | c = Effect of Elementary Row Operations on Determinant }} {{eqn | r = \paren {-c} \map \det {\mathbf A''}       | c = Product with Ring Negative: Corollary }} {{end-eqn}} Then the induction step is proved for all three types of elementary row operations. {{qed}} Category:Determinants Category:Elementary Row Operations Category:Proofs by Induction lxp6dfv6nef3s6nkzxs4nrdn7e7j5vo