Effect of Sequence of Elementary Row Operations on Determinant

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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:

\(\displaystyle \det \left({\mathbf A}\right)\) \(=\) \(\displaystyle c \det \left({\mathbf A'}\right)\) by the induction hypothesis
\(\displaystyle \) \(=\) \(\displaystyle \left({ca}\right) \det \left({\mathbf A''}\right)\) by Effect of Elementary Row Operations on Determinant


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:

\(\displaystyle \det \left({\mathbf A}\right)\) \(=\) \(\displaystyle c \det \left({\mathbf A'}\right)\) by the induction hypothesis
\(\displaystyle \) \(=\) \(\displaystyle c1_R \det \left({\mathbf A''}\right)\) by Effect of Elementary Row Operations on Determinant
\(\displaystyle \) \(=\) \(\displaystyle c \det \left({\mathbf A''}\right)\) by definition of identity element, since $R$ is commutative


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

Then:

\(\displaystyle \det \left({\mathbf A}\right)\) \(=\) \(\displaystyle c \det \left({\mathbf A'}\right)\) by the induction hypothesis
\(\displaystyle \) \(=\) \(\displaystyle c \left({-1_R}\right) \det \left({\mathbf A''}\right)\) by Effect of Elementary Row Operations on Determinant
\(\displaystyle \) \(=\) \(\displaystyle \left({-c}\right) \det \left({\mathbf A''}\right)\) by Product with Ring Negative/Corollary


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

$\blacksquare$