Effect of Sequence of Elementary Row Operations on Determinant

Theorem
Let $\hat o_1, \ldots, \hat o_m$ be a finite sequence of elementary row operations.

Here, $\hat o_i$ denotes an elementary row operation on a square matrix of order $n$ over a commutative ring with unity $\left({R, +, \circ}\right)$.

Here, $i \in \left\{ {1, \ldots, m}\right\}$.

Then there exists $c \in R$ such that for all $n \times n$-matrices $\mathbf A$ over $R$:


 * $\det \left({\mathbf A}\right) = c \det \left({\mathbf A'}\right)$

where $\mathbf A'$ is the $n \times n$-matrix that results from using the elementary row operations $\hat o_1, \ldots, \hat o_m$ on $\mathbf A$.

Proof
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 $r_i \to ar_i$, where $a \in R$ and $i \in \left\{ {1, \ldots, n}\right\}$.

From Effect of Elementary Row Operations on Determinant, it follows that


 * $\det \left({\mathbf A}\right) = a \det \left({\mathbf A'}\right)$

Suppose that $\hat o$ is of the type $r_i \to r_i + ar_j$, where $a \in R$ and $i,j \in \left\{ {1, \ldots, n}\right\}, i \ne j$.

From Effect of Elementary Row Operations on Determinant, it follows that


 * $\det \left({\mathbf A}\right) = \det \left({\mathbf A'}\right) = 1_R \det \left({\mathbf A'}\right)$

where $1_R$ denotes the identity element of $\left({R, \circ}\right)$.

Suppose that $\hat o$ is of the type $r_i \leftrightarrow r_j$.

From Effect of Elementary Row Operations on Determinant, it follows that


 * $\det \left({\mathbf A}\right) = -\det \left({\mathbf A'}\right) = -1_R \det \left({\mathbf A'}\right)$

where the last equality follows from Product with Ring Negative/Corollary.

Then the induction basis is proved for all three types of elementary row operations.

Induction Hypothesis
For $m \in \N$, let $\hat o_1, \ldots, \hat o_m$ be a finite sequence of elementary row operation.

Then, the induction hypothesis is:

There exists $c \in R$ such that for all $n \times n$-matrices $\mathbf A$:


 * $\det \left({\mathbf A}\right) = c \det \left({\mathbf A'}\right)$

where $\mathbf A'$ is the $n \times n$-matrix that results from using the elementary row operations $\hat o_1, \ldots, \hat o_m$ on $\mathbf A$.

Induction Step
Let $\hat o_1, \ldots, \hat o_m, \hat o_{m+1}$ be a finite sequence of elementary row operation.

Let $r_i$ denote the $i$'th row of $\mathbf A'$.

Let $\mathbf A''$ denote the matrix $n \times n$-matrix 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 $i \in \left\{ {1, \ldots, n}\right\}$.

Then:

Suppose that $\hat o_{m+1}$ is of the type $r_i \to r_i + ar_j$, where $a \in R$ and $i,j \in \left\{ {1, \ldots, n}\right\}, i \ne j$.

Then:

Suppose that $\hat o_{m+1}$ is of the type $r_i \leftrightarrow r_j$.

Then:

Then the induction step is proved for all three types of elementary row operations.