Linear AlgebraChallengeRow Reduction & Rank

Row Reduction & Rank · Challenge problem

For how many integers $k$ with $-10 \le k \le 10$ does elimination without row exchanges on

A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & k & 2 \\ 1 & 2 & k \end{pmatrix}

produce three nonzero pivots (so that $A = LU$ with $U$ invertible)?