Inverse of a 0-1 matrix

Task number: 2428

Invert the matrix

\(\mathbf A=\begin{pmatrix} 1&1&1&\cdots&1\\ 0&1&1&\cdots&1\\ 0&0&1&\cdots&1\\ \vdots&&&\ddots&\vdots\\ 0&0&0&\cdots&1 \end{pmatrix}\)

  • Resolution

    \[ \left( \begin{matrix} 1 & 1 & 1 & \cdots & 1 \\ 0 & 1 & 1 & \cdots & 1 \\ 0 & 0 & 1 & \cdots & 1 \\ \vdots & & & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{matrix}\right|\left. \begin{matrix} 1&0&0&\cdots&0\\0&1&0&\cdots&0\\0&0&1&\cdots&0\\\vdots&&&\ddots&\vdots\\0&0&0&\cdots&1 \end{matrix} \right) \sim \left( \begin{matrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 1 & \cdots & 1 \\ 0 & 0 & 1 & \cdots & 1 \\ \vdots & & & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{matrix}\right|\left. \begin{matrix} 1&-1&0&\cdots&0\\0&1&0&\cdots&0\\0&0&1&\cdots&0\\\vdots&&&\ddots&\vdots\\0&0&0&\cdots&1 \end{matrix} \right) \] \[ \sim…\sim \left( \begin{matrix} 1&0&0&\cdots&0\\0&1&0&\cdots&0\\0&0&1&\cdots&0\\\vdots&&&\ddots&\vdots\\0&0&0&\cdots&1 \end{matrix}\right|\left. \begin{matrix} 1&-1&0&\cdots&0\\0&1&-1&\cdots&0\\0&0&1&\cdots&0\\\vdots&&&\ddots&\vdots\\0&0&0&\cdots&1 \end{matrix} \right) \]
  • Result

    The inverse is

    \(\mathbf A^{-1}=\begin{pmatrix} 1&-1&0&\cdots&0\\ 0&1&-1&\ddots&\vdots\\ 0&0&1&\ddots&0\\ \vdots&\vdots&\ddots&\ddots&-1\\ 0&0&\cdots&0&1 \end{pmatrix}\)

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