# Determinants of Elementary Matrices

Recall that an elementary matrix $E$ arises from performing exactly one of the following elementary row operations on $I$:

**1.**Multiplying a row by a constant $k$ where $k ≠ 0$ ($kR_{a} \to R_{a}$).

**2.**Adding (or subtracting) a multiple $k$ of a row to another ($R_a + kR_b \to R_a$).

**3.**Interchanging two rows ($R_a \leftrightarrow R_b$).

We will now look at some techniques in evaluating the determinants of these elementary matrices.

## Determinants of Elementary Matrices from Multiplying a Row by a Constant

If $E$ results from multiplying a single row of $I$ by a constant $k$, it follows that $\det(E) = k$. For example, consider the following elementary matrix has $\det(E) = 5$.

(1)

## Determinants of Elementary Matrices by Adding/Subtracting a Multiple of One Row to Another

If $E$ results from adding or subtracting a multiple of one row of $I$ to another, then it follows that $\det(E) = 1$. For example, the following elementary matrix has $\det(E) = 1$.

(2)

## Determinants of Elementary Matrices by Interchanging Two Rows

If $E$ results from interchanging two rows of $I$, then it follows that $\det(E) = -1$. For example, the following elementary matrix has $\det(E) = -1$.

(3)

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