# Determinante de $\left[\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right]$

La calculadora encontrará el determinante de la matriz cuadrada $3$ x $3$ $\left[\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right]$, con los pasos que se muestran.

Si la calculadora no calculó algo o ha identificado un error, o tiene una sugerencia/comentario, escríbalo en los comentarios a continuación.

### Tu aportación

Calcular $\left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right|$.

### Solución

Reste la columna $1$ multiplicada por $\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$ de la columna $2$: $C_{2} = C_{2} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} C_{1}$.

$\left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right| = \left|\begin{array}{ccc}\cos{\left(x \right)} & 0 & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & 0 & - 4 \sin{\left(2 x \right)}\end{array}\right|$

Reste la columna $1$ multiplicada por $\frac{\sin{\left(2 x \right)}}{\cos{\left(x \right)}}$ de la columna $3$: $C_{3} = C_{3} - \frac{\sin{\left(2 x \right)}}{\cos{\left(x \right)}} C_{1}$.

$\left|\begin{array}{ccc}\cos{\left(x \right)} & 0 & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & 0 & - 4 \sin{\left(2 x \right)}\end{array}\right| = \left|\begin{array}{ccc}\cos{\left(x \right)} & 0 & 0\\- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\- \cos{\left(x \right)} & 0 & - 3 \sin{\left(2 x \right)}\end{array}\right|$

Expanda a lo largo de la fila $1$:

$\left|\begin{array}{ccc}\cos{\left(x \right)} & 0 & 0\\- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\- \cos{\left(x \right)} & 0 & - 3 \sin{\left(2 x \right)}\end{array}\right| = \left(\cos{\left(x \right)}\right) \left(-1\right)^{1 + 1} \left|\begin{array}{cc}\frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\0 & - 3 \sin{\left(2 x \right)}\end{array}\right| + \left(0\right) \left(-1\right)^{1 + 2} \left|\begin{array}{cc}- \sin{\left(x \right)} & 2 \cos^{2}{\left(x \right)}\\- \cos{\left(x \right)} & - 3 \sin{\left(2 x \right)}\end{array}\right| + \left(0\right) \left(-1\right)^{1 + 3} \left|\begin{array}{cc}- \sin{\left(x \right)} & \frac{1}{\cos{\left(x \right)}}\\- \cos{\left(x \right)} & 0\end{array}\right| = \cos{\left(x \right)} \left|\begin{array}{cc}\frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\0 & - 3 \sin{\left(2 x \right)}\end{array}\right|$

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

$\left|\begin{array}{cc}\frac{1}{\cos{\left(x \right)}} & 2 \cos^{2}{\left(x \right)}\\0 & - 3 \sin{\left(2 x \right)}\end{array}\right| = \left(\frac{1}{\cos{\left(x \right)}}\right)\cdot \left(- 3 \sin{\left(2 x \right)}\right) - \left(2 \cos^{2}{\left(x \right)}\right)\cdot \left(0\right) = - 6 \sin{\left(x \right)}$

Finalmente, $\left(\cos{\left(x \right)}\right)\cdot \left(- 6 \sin{\left(x \right)}\right) = - 3 \sin{\left(2 x \right)}$.

### Respuesta

$\left|\begin{array}{ccc}\cos{\left(x \right)} & \sin{\left(x \right)} & \sin{\left(2 x \right)}\\- \sin{\left(x \right)} & \cos{\left(x \right)} & 2 \cos{\left(2 x \right)}\\- \cos{\left(x \right)} & - \sin{\left(x \right)} & - 4 \sin{\left(2 x \right)}\end{array}\right| = - 3 \sin{\left(2 x \right)}$A