Determinant of Elementary Row Matrix/Scale Row and Add

Theorem

Let $e_2$ be the elementary row operation $\text {ERO} 2$:

$(\text {ERO} 2)$

$:$

$\ds r_i \to r_i + \lambda r_j $

For some $\lambda$, add $\lambda$ times row $j$ to row $i$

which is to operate on some arbitrary matrix space.

Let $\mathbf E_2$ be the elementary row matrix corresponding to $e_2$.

The determinant of $\mathbf E_2$ is:

$\map \det {\mathbf E_2} = 1$

$E_{a b} = \delta_{a b} + \lambda \cdot \delta_{a i} \cdot \delta_{j b}$

where:

$E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$

$\delta_{a b}$ is the Kronecker delta:

$\delta_{a b} = \begin {cases} 1 & : \text {if $a = b$} \\ 0 & : \text {if $a \ne b$} \end {cases}$

Because $i \ne j$ it follows that:

if $a = i$ and $b = j$ then $a \ne b$

Hence when $a = b$ we have that:

$\delta_{a i} \cdot \delta_{j b} = 0$

Hence the diagonal elements of $\mathbf E_2$ are all equal to $1$.

We also have that $\delta_{a i} \cdot \delta_{j b} = 1$ if and only if $a = i$ and $b = j$.

Hence, all elements of $\mathbf E_2$ apart from the diagonal elements and $a_{i j}$ are equal to $0$.

Thus $\mathbf E_2$ is a triangular matrix (either upper or lower).

From Determinant of Triangular Matrix, $\map \det {\mathbf E_2}$ is equal to the product of all the diagonal elements of $\mathbf E_2$.

But as we have seen, these are all equal to $1$.

Hence the result.

$\blacksquare$

1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.6$ Determinant and trace: Fact $1.8 \ \text {(c)}$