Calcular determinantes

Calcula los determinantes de las siguientes matrices:

$A = \left( \begin{array}{cccc} 1 & 2 & -1 & 2 \\ 2 & 4 & 0 & -2 \\ -1 & 3 & 0 & 1 \\ -3 & 2 & 0 & 4 \end{array} \right)$ $\qquad$

$B = \left( \begin{array}{cccc} 2 & 0 & -1 & -2 \\ 0 & 4 & 2 & -1 \\ -1 & 3 & 0 & -1 \\ -3 & 0 & 5 & 3 \end{array} \right)$

SOLUCIÓN

Determinante 4×4 — Desarrollo por adjuntos

$\left|\begin{array}{cccc}1 & 2 & -1 & 2 \\ 2 & 4 & 0 & -2 \\ -1 & 3 & 0 & 1 \\ -3 & 2 & 0 & 4\end{array}\right|$

Desarrollamos por la columna 3 (tiene 3 ceros):

$\left|\begin{array}{cccc}1 & 2 & {\color[RGB]{0,0,0}{-1}} & 2 \\ 2 & 4 & {\color[RGB]{30,100,220}{0}} & -2 \\ -1 & 3 & {\color[RGB]{200,30,30}{0}} & 1 \\ -3 & 2 & {\color[RGB]{0,155,50}{0}} & 4\end{array}\right|$

${\color[RGB]{0,0,0}{(-1)}}\cdot(-1)^{1+3}\cdot\left|\begin{array}{ccc}2 & 4 & -2 \\ -1 & 3 & 1 \\ -3 & 2 & 4\end{array}\right|={\color[RGB]{0,0,0}{(-1)}}\cdot10={\color[RGB]{0,0,0}{-10}}$

$\det={\color[RGB]{0,0,0}{-10}}=\boxed{-10}$

Determinante 4×4 — Desarrollo por adjuntos

$\left|\begin{array}{cccc}2 & 0 & -1 & -2 \\ 0 & 4 & 2 & -1 \\ -1 & 3 & 0 & -1 \\ -3 & 0 & 5 & 3\end{array}\right|$

Desarrollamos por la columna 2 (tiene 2 ceros):

$\left|\begin{array}{cccc}2 & {\color[RGB]{0,0,0}{0}} & -1 & -2 \\ 0 & {\color[RGB]{30,100,220}{4}} & 2 & -1 \\ -1 & {\color[RGB]{200,30,30}{3}} & 0 & -1 \\ -3 & {\color[RGB]{0,155,50}{0}} & 5 & 3\end{array}\right|$

${\color[RGB]{30,100,220}{4}}\cdot(-1)^{2+2}\cdot\left|\begin{array}{ccc}2 & -1 & -2 \\ -1 & 0 & -1 \\ -3 & 5 & 3\end{array}\right|={\color[RGB]{30,100,220}{4}}\cdot14={\color[RGB]{30,100,220}{56}}$

${\color[RGB]{200,30,30}{3}}\cdot(-1)^{3+2}\cdot\left|\begin{array}{ccc}2 & -1 & -2 \\ 0 & 2 & -1 \\ -3 & 5 & 3\end{array}\right|={\color[RGB]{200,30,30}{3}}\cdot(-7)={\color[RGB]{200,30,30}{-21}}$

$\det={\color[RGB]{30,100,220}{56}}-{\color[RGB]{200,30,30}{21}}=\boxed{35}$

Comentar el ejercicio