First slide

Introduction to Determinants

Question

If y = sin mx, then the value of the determinant $|\begin{matrix} y & y_{1} & y_{2} \\ y_{3} & y_{4} & y_{5} \\ y_{6} & y_{7} & y_{8} \end{matrix}|$ , where $y_{n} = \frac{d^{n} y}{{dx}^{n}}$ is

Moderate

A

m⁹
B

m²
C

m³
D

0

Solution

We have y = sin mx, therefore y₁ = m cos mx, y₂= -m² sin mx, etc.
$\begin{matrix} ∴ Δ = |\begin{array}{ccc} y & y_{1} & y_{2} \\ y_{3} & y_{4} & y_{5} \\ y_{6} & y_{7} & y_{8} \end{array}| \\ = |\begin{array}{ccc} \sin mx & mcos mx & - m^{2} \sin mx \\ - m^{3} \cos mx & m^{4} \sin mx & m^{5} \cos mx \\ - m^{6} \sin mx & - m^{7} \cos mx & m^{8} \sin mx \end{array}| \\ = m^{12} |\begin{array}{ccc} \sin mx & \cos mx & - \sin mx \\ - \cos mx & \sin mx & \cos mx \\ - \sin mx & - \cos mx & \sin mx \end{array}| = 0 \end{matrix}$

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