Трећи разред средње школе

Детерминанте - примери 2

Решавање једначина задатих у облику детерминанте.

Задаци

Текст задатака објашњених у видео лекцији.

Пр.1)   Решити једначину у облику детерминанте.

            $\left| {\begin{array}{*{20}{c}}
  a&a&a \\
  { - a}&a&x \\
  { - a}&{ - a}&x
\end{array}} \right| = 0$

Пр.2)   Решити једначину задату преко детерминанте.

            $\left| {\begin{array}{*{20}{c}}
  1&1&1 \\
  1&x&x \\
  1&{{x^2}}&x
\end{array}} \right| = 0$

Пр.1)

\[\begin{gathered}
\left| {\begin{array}{*{20}{c}}
a&a&a \\
{ - a}&a&x \\
{ - a}&{ - a}&x
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
&& \\
0&{2}&{x + a} \\
0&0&{x + a}
\end{array}} \right| = a \cdot \left| {\begin{array}{*{20}{c}}
{2a}&{x + a} \\
0&{x + a}
\end{array}} \right| = \hfill \\
= a\left( {x + a} \right)\left| {\begin{array}{*{20}{c}}
{2a}&1 \\
0&1
\end{array}} \right| = a\left( {x + a} \right)\left( {2a - 0} \right) = 2{a^2}\left( {x + a} \right) \hfill \\
\hfill \\
2{a^2}\left( {x + a} \right) = 0 \hfill \\
x + a = 0 \hfill \\
x = - a \hfill \\
\end{gathered} \]

Пр.2)  

\[\begin{gathered}
\left| {\begin{array}{*{20}{c}}
1&1&1 \\
1&x&x \\
1&{{x^2}}&x
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
1&0&0 \\
1&{x - 1}&{x - 1} \\
1&{{x^2} - 1}&{x - 1}
\end{array}} \right| = 1\left| {\begin{array}{*{20}{c}}
{x - 1}&{x - 1} \\
{{x^2} - 1}&{x - 1}
\end{array}} \right| = \hfill \\
\hfill \\
= \left( {x - 1} \right)\left| {\begin{array}{*{20}{c}}
1&1 \\
{\left( {x - 1} \right)\left( {x + 1} \right)}&{\left( {x - 1} \right)}
\end{array}} \right| = {\left( {x - 1} \right)^2}\left| {\begin{array}{*{20}{c}}
1&1 \\
{\left( {x + 1} \right)}&1
\end{array}} \right| = \hfill \\
\hfill \\
= {\left( {x - 1} \right)^2}\left( {1 - \left( {x + 1} \right)} \right) = {\left( {x - 1} \right)^2}\left( {1 - x - 1} \right) = {\left( {x - 1} \right)^2}\left( { - x} \right) \hfill \\
\hfill \\
{\left( {x - 1} \right)^2}\left( { - x} \right) = 0 \hfill \\
\begin{array}{*{20}{c}}
{x - 1 = 0}&{ - x = 0} \\
{x = 1}&{x = 0}
\end{array} \hfill \\
\end{gathered} \]