# Elementary matrices are invertible/nonsingular.

Put content here**Theorem.** Every elementary matrix is invertible (nonsingular), and its inverse is also an elementary matrix of the same type.
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Specifically:
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1. **Row interchange:** $E_{ij}^{-1} = E_{ij}$ (self-inverse)
2. **Row scaling:** $E_i(c)^{-1} = E_i(1/c)$ for $c \neq 0$
3. **Row replacement:** $E_{ij}(c)^{-1} = E_{ij}(-c)$
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**Proof.** Each elementary row operation can be reversed by another elementary row operation. The matrix that reverses the operation is the inverse.
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**Example.** If $E = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 2 & 0 & 1 \end{pmatrix}$ adds 2 times row 1 to row 3, then $E^{-1} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ -2 & 0 & 1 \end{pmatrix}$ subtracts 2 times row 1 from row 3.

# Parents

* Elementary matrices