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.