Inverses of elementary matrices

Task number: 2426

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.

Difficulty level: Easy task (using definitions and simple reasoning)
Solution require uncommon idea
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