Without expanding, show that the value of the following determinant is zero: \[\begin{vmatrix}\sin\alpha & \cos\alpha & \cos(\alpha + \delta) \\ \sin\beta & \cos\beta & \cos(\beta + \delta) \\ \sin\gamma & \cos\gamma & \cos(\gamma + \delta)\end{vmatrix}\]

\[\begin{vmatrix}\sin\alpha & \cos\alpha & \cos(\alpha + \delta) \\ \sin\beta & \cos\beta & \cos(\beta + \delta) \\ \sin\gamma & \cos\gamma & \cos(\gamma + \delta)\end{vmatrix}\] \[ = \begin{vmatrix}\sin\alpha\sin\delta & \cos\alpha\cos\delta & \cos(\alpha + \delta) \\ \sin\beta\sin\delta & \cos\beta\cos\delta & \cos(\beta + \delta) \\ \sin\gamma\sin\delta & \cos\gamma\cos\delta & \cos(\gamma + \delta)\end{vmatrix} \left[\text{ Applying }C_1 \to \sin\delta C_1 \text{ and }C_2 \to \cos\delta C_2 \right]\] \[ = \begin{vmatrix}\sin\alpha\sin\delta & \cos(\alpha + \delta) & \cos(\alpha + \delta) \\ \sin\beta\sin\delta & \cos(\beta + \delta) & \cos(\beta + \delta) \\ \sin\gamma\sin\delta & \cos(\gamma + \delta) & \cos(\gamma + \delta)\end{vmatrix} \left[ \text{ Applying }C_2 \to C_2 - C_1 \right]\] \[ = 0\]

Without Expanding, Show that the Value of the Following Determinant is Zero: ∣ ∣ ∣ ∣ ∣ Sin α Cos α Cos ( α + δ ) Sin β Cos β Cos ( β + δ ) Sin γ Cos γ Cos ( γ + δ ) ∣ ∣ ∣ ∣ ∣ - Mathematics

Question

Without expanding, show that the value of the following determinant is zero:

$| \begin{matrix} \sin α & \cos α & \cos (α + δ) \\ \sin β & \cos β & \cos (β + δ) \\ \sin γ & \cos γ & \cos (γ + δ) \end{matrix} |$

Solution

$| \begin{matrix} \sin α & \cos α & \cos (α + δ) \\ \sin β & \cos β & \cos (β + δ) \\ \sin γ & \cos γ & \cos (γ + δ) \end{matrix} |$
$= | \begin{matrix} \sin α \sin δ & \cos α \cos δ & \cos (α + δ) \\ \sin β \sin δ & \cos β \cos δ & \cos (β + δ) \\ \sin γ \sin δ & \cos γ \cos δ & \cos (γ + δ) \end{matrix} | [Applying C_{1} \to \sin δ C_{1} and C_{2} \to \cos δ C_{2}]$
$= | \begin{matrix} \sin α \sin δ & \cos (α + δ) & \cos (α + δ) \\ \sin β \sin δ & \cos (β + δ) & \cos (β + δ) \\ \sin γ \sin δ & \cos (γ + δ) & \cos (γ + δ) \end{matrix} | [Applying C_{2} \to C_{2} - C_{1}]$
$= 0$

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Applications of Determinants and Matrices

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Q 2.13 Q 2.12 Q 2.14

Chapter 6: Determinants - Exercise 6.2 [Page 57]

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Chapter 6 Determinants
Exercise 6.2 | Q 2.13 | Page 57

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Without Expanding, Show that the Value of the Following Determinant is Zero: ∣ ∣ ∣ ∣ ∣ Sin α Cos α Cos ( α + δ ) Sin β Cos β Cos ( β + δ ) Sin γ Cos γ Cos ( γ + δ ) ∣ ∣ ∣ ∣ ∣ - Mathematics

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