Question

Quais os valores de k para que o determinante da matriz $\left[\begin{array}{ccc}4&2&-k\\-1&k&1\\k&0&k\end{array}\right]$ seja nulo?

Answer 1

$Encontando o determinante : \\\\\left| \begin{array}{ccccc} 4 & 2 & -k & 4 & 2 \\ -1 & k & 1 & -1 & k \\ k & 0 & k & k & 0 \end{array} \right|\\\\det=(4k^2+2k+0)-(-k^3+0-2k)\\det=4k^2+2k+k^3+2k\\det=k^3+4k^2+4k\\\\Para que o determinante seja nulo : \\\\k^3+4k^2+4k=0\\k(k^2+4k+4)=0\\k'=0\\\\k^2+4k+4=0\\\\\Delta=(4)^2-4\cdot(1)\cdot(4)\\\Delta=16-16\\\Delta=0\\\\k=\frac{-4\pm0}{2}\\\\k''=\frac{-4-0}{2}=-2\\\\s=\{k=0, k=-2\}$