Show that $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}=LU$ is impossible where $L$ is lower triangular and $U$ is upper triangular.
2026-03-25 05:58:54.1774418334
Showing $LU$ is impossible...
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We have $$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} = \begin{bmatrix} a & 0 \\ b & c \end{bmatrix}\begin{bmatrix} x & y \\ 0 & z \end{bmatrix} = \begin{bmatrix} ax & ay \\ bx & by+cz \end{bmatrix}.$$ From $bx = ay = 1$ it follows that $a,x\ne 0$ but this contradicts $ax = 0$.