Calcular determinantes

determinantes Ejercicios_Resueltos matrices

Calcula los determinantes de las siguientes matrices
$A = \left( \begin{array}{cc} 1 & -8 \\ 0 & 3 \end{array} \right)$
$\qquad$ $B= \left( \begin{array}{cc} 3 & -1 \\ -15 & -4 \end{array} \right)$

$C = \left( \begin{array}{ccc} 1 & -2 & 1 \\ 0 & -2 & 4 \\ 1 & 3 & 5 \end{array} \right)$
$\qquad$ $D = \left( \begin{array}{ccc} 5 & -1 & 2 \\ 1 & 2 & 3 \\ 6 & 1 & 5 \end{array} \right)$

SOLUCIÓN

Determinante 2×2

$\left|\begin{array}{cc}1 & -8 \\ 0 & 3\end{array}\right|$

$= 1\cdot3 - (-8)\cdot0$

$= \boxed{3}$

Determinante 2×2

$\left|\begin{array}{cc}3 & -1 \\ -15 & -4\end{array}\right|$

$= 3\cdot(-4) - (-1)\cdot(-15)$

$= \boxed{-27}$

Determinante 3×3 — Regla de Sarrus

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

Diagonales principales ↘ (productos positivos):

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

${\color[RGB]{0,0,0}{1}}{\cdot}{\color[RGB]{0,0,0}{(-2)}}{\cdot}{\color[RGB]{0,0,0}{5}}+{\color[RGB]{30,100,220}{(-2)}}{\cdot}{\color[RGB]{30,100,220}{4}}{\cdot}{\color[RGB]{30,100,220}{1}}+{\color[RGB]{200,30,30}{1}}{\cdot}{\color[RGB]{200,30,30}{0}}{\cdot}{\color[RGB]{200,30,30}{3}}={\color[RGB]{0,0,0}{(-10)}}+{\color[RGB]{30,100,220}{(-8)}}+{\color[RGB]{200,30,30}{0}}=-18$

Diagonales secundarias ↗ (productos negativos):

$\left|\begin{array}{ccc}{\color[RGB]{0,155,50}{1}} & {\color[RGB]{110,0,200}{-2}} & {\color[RGB]{0,0,0}{1}} \\ {\color[RGB]{110,0,200}{0}} & {\color[RGB]{0,0,0}{-2}} & {\color[RGB]{0,155,50}{4}} \\ {\color[RGB]{0,0,0}{1}} & {\color[RGB]{0,155,50}{3}} & {\color[RGB]{110,0,200}{5}}\end{array}\right|$

${\color[RGB]{0,0,0}{1}}{\cdot}{\color[RGB]{0,0,0}{(-2)}}{\cdot}{\color[RGB]{0,0,0}{1}}+{\color[RGB]{0,155,50}{1}}{\cdot}{\color[RGB]{0,155,50}{4}}{\cdot}{\color[RGB]{0,155,50}{3}}+{\color[RGB]{110,0,200}{(-2)}}{\cdot}{\color[RGB]{110,0,200}{0}}{\cdot}{\color[RGB]{110,0,200}{5}}={\color[RGB]{0,0,0}{(-2)}}+{\color[RGB]{0,155,50}{12}}+{\color[RGB]{110,0,200}{0}}=10$

$\det=-18-10=\boxed{-28}$

Determinante 3×3 — Regla de Sarrus

$\left|\begin{array}{ccc}5 & -1 & 2 \\ 1 & 2 & 3 \\ 6 & 1 & 5\end{array}\right|$

Diagonales principales ↘ (productos positivos):

$\left|\begin{array}{ccc}{\color[RGB]{0,0,0}{5}} & {\color[RGB]{30,100,220}{-1}} & {\color[RGB]{200,30,30}{2}} \\ {\color[RGB]{200,30,30}{1}} & {\color[RGB]{0,0,0}{2}} & {\color[RGB]{30,100,220}{3}} \\ {\color[RGB]{30,100,220}{6}} & {\color[RGB]{200,30,30}{1}} & {\color[RGB]{0,0,0}{5}}\end{array}\right|$

${\color[RGB]{0,0,0}{5}}{\cdot}{\color[RGB]{0,0,0}{2}}{\cdot}{\color[RGB]{0,0,0}{5}}+{\color[RGB]{30,100,220}{(-1)}}{\cdot}{\color[RGB]{30,100,220}{3}}{\cdot}{\color[RGB]{30,100,220}{6}}+{\color[RGB]{200,30,30}{2}}{\cdot}{\color[RGB]{200,30,30}{1}}{\cdot}{\color[RGB]{200,30,30}{1}}={\color[RGB]{0,0,0}{50}}+{\color[RGB]{30,100,220}{(-18)}}+{\color[RGB]{200,30,30}{2}}=34$

Diagonales secundarias ↗ (productos negativos):

$\left|\begin{array}{ccc}{\color[RGB]{0,155,50}{5}} & {\color[RGB]{110,0,200}{-1}} & {\color[RGB]{0,0,0}{2}} \\ {\color[RGB]{110,0,200}{1}} & {\color[RGB]{0,0,0}{2}} & {\color[RGB]{0,155,50}{3}} \\ {\color[RGB]{0,0,0}{6}} & {\color[RGB]{0,155,50}{1}} & {\color[RGB]{110,0,200}{5}}\end{array}\right|$

${\color[RGB]{0,0,0}{2}}{\cdot}{\color[RGB]{0,0,0}{2}}{\cdot}{\color[RGB]{0,0,0}{6}}+{\color[RGB]{0,155,50}{5}}{\cdot}{\color[RGB]{0,155,50}{3}}{\cdot}{\color[RGB]{0,155,50}{1}}+{\color[RGB]{110,0,200}{(-1)}}{\cdot}{\color[RGB]{110,0,200}{1}}{\cdot}{\color[RGB]{110,0,200}{5}}={\color[RGB]{0,0,0}{24}}+{\color[RGB]{0,155,50}{15}}+{\color[RGB]{110,0,200}{(-5)}}=34$

$\det=34-34=\boxed{0}$

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