## Inverses of elementary matrices

Determine inverses to the following elementary matrices:

• #### Variant 1

$$E_{i,j} = \begin{pmatrix} 1 & 0 & \cdots & & & \cdots & 0 \\ 0 & \ddots & \ddots & & & & \vdots \\ \vdots & \ddots & 0 & & 1 & & \\ & & & \ddots & & \\ & & 1 & & 0 & \ddots & \vdots \\ \vdots & & & & \ddots & \ddots & 0 \\ 0 & \cdots & & & \cdots & 0 & 1 \end{pmatrix}$$

This matrix is obtained from the unit matrix by swapping the $$i$$-th and the $$j$$-th rows.

• #### Variant 2

$$E_i(m) = \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0\\ 0 & \ddots & \ddots & & \vdots \\ \vdots & \ddots & m & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & \cdots & \cdots & 0 & 1 \end{pmatrix}$$

$$m$$ appears only at the $$i$$-th column and $$i$$-th row, and $$m\ne 0$$

• #### Variant 3

$$E_{i,j}(m) = \begin{pmatrix} 1 & 0 & \cdots & \cdots & 0 \\ 0 & \ddots & \ddots & & \vdots & \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & m & \ddots & \ddots & 0 \\ 0 & \cdots & \cdots & 0 & 1 \end{pmatrix}$$

$$m$$ appears only at the $$i$$-th column and $$j$$-th row.