Determinante de $$$\left[\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-3 & 2 & 1\end{array}\right]$$$

Calcular $$$\left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-3 & 2 & 1\end{array}\right|$$$.

Reste la fila $$$1$$$ de la fila $$$3$$$: $$$R_{3} = R_{3} - R_{1}$$$.

$$$\left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-3 & 2 & 1\end{array}\right| = \left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-1 & -1 & 0\end{array}\right|$$$

Expanda a lo largo de la columna $$$3$$$:

$$$\left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-1 & -1 & 0\end{array}\right| = \left(1\right) \left(-1\right)^{1 + 3} \left|\begin{array}{cc}7 & -4\\-1 & -1\end{array}\right| + \left(0\right) \left(-1\right)^{2 + 3} \left|\begin{array}{cc}-2 & 3\\-1 & -1\end{array}\right| + \left(0\right) \left(-1\right)^{3 + 3} \left|\begin{array}{cc}-2 & 3\\7 & -4\end{array}\right| = \left|\begin{array}{cc}7 & -4\\-1 & -1\end{array}\right|$$$

El determinante de una matriz de 2x2 es $$$\left|\begin{array}{cc}a & b\\c & d\end{array}\right| = a d - b c$$$.

$$$\left|\begin{array}{cc}7 & -4\\-1 & -1\end{array}\right| = \left(7\right)\cdot \left(-1\right) - \left(-4\right)\cdot \left(-1\right) = -11$$$

$$$\left|\begin{array}{ccc}-2 & 3 & 1\\7 & -4 & 0\\-3 & 2 & 1\end{array}\right| = -11$$$A