Детерминанте – примери 1

Текст задатака објашњених у видео лекцији.

Пр.1)   Решити следећу детерминанту.

\[\left| {\begin{array}{*{20}{c}}
3&{ - 7} \\
2&1
\end{array}} \right| = 3 \cdot 1 - 2 \cdot \left( { - 7} \right) = 3 + 14 = 17\]

Пр.2)   

\[\begin{gathered}
\left| {\begin{array}{*{20}{c}}
{12}&{ - 9} \\
8&1
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
{4 \cdot 3}&{ - 9} \\
{4 \cdot 2}&1
\end{array}} \right| = 4\left| {\begin{array}{*{20}{c}}
{3 \cdot 1}&{3 \cdot \left( { - 3} \right)} \\
2&1
\end{array}} \right| = \hfill \\
\hfill \\
= 4 \cdot 3\left| {\begin{array}{*{20}{c}}
1&{ - 3} \\
2&1
\end{array}} \right| = 12 \cdot \left( {1 + 6} \right) = 12 \cdot 7 = 84 \hfill \\
\end{gathered} \]

Пр.3)   

\[\begin{gathered}
\left| {\begin{array}{*{20}{c}}
{57}&{29} \\
{54}&{28}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
{3 \cdot 19}&{29} \\
{3 \cdot 18}&{28}
\end{array}} \right| = 3 \cdot \left| {\begin{array}{*{20}{c}}
{19}&{29} \\
{18}&{28}
\end{array}} \right| = \hfill \\
\hfill \\
= 3\left| {\begin{array}{*{20}{c}}
{19 - 18}&{29 - 18} \\
{18}&{28}
\end{array}} \right| = 3 \cdot \left| {\begin{array}{*{20}{c}}
1&1 \\
{18}&{28}
\end{array}} \right| = 3\left( {28 - 18} \right) = 30 \hfill \\
\end{gathered} \]

Пр.4)   

\[\begin{gathered}
\left| {\begin{array}{*{20}{c}}
{108}&{ - 4} \\
{ - 54}&2
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
{27 \cdot 4}&{ - 4} \\
{27\left( { - 2} \right)}&2
\end{array}} \right| = 27\left| {\begin{array}{*{20}{c}}
4&{ - 4} \\
{ - 2}&2
\end{array}} \right| = \hfill \\
\hfill \\
= 27 \cdot 4 \cdot 2\left| {\begin{array}{*{20}{c}}
1&{ - 1} \\
{ - 1}&1
\end{array}} \right| = 27 \cdot 4 \cdot 2 \cdot \left( {1 - 1} \right) = 0 \hfill \\
\end{gathered} \]

Пр.5)

\[\begin{gathered}
\left| {\begin{array}{*{20}{c}}
x&{x + 1} \\
{ - 4}&{3x + 3}
\end{array}} \right| = 0 \hfill \\
\hfill \\
\left| {\begin{array}{*{20}{c}}
x&{x + 1} \\
{ - 4}&{3\left( {x + 1} \right)}
\end{array}} \right| = 0 \hfill \\
\hfill \\
\left( {x + 1} \right)\left| {\begin{array}{*{20}{c}}
x&1 \\
{ - 4}&3
\end{array}} \right| = 0 \hfill \\
\hfill \\
\left( {x + 1} \right)\left( {3x + 4} \right) = 0 \hfill \\
\hfill \\
\begin{array}{*{20}{c}}
{x + 1 = 0}&{3x + 4 = 0} \\
{x = - 1}&{x = - \frac{4}{3}}
\end{array} \hfill \\
\end{gathered} \]