Matrices & Determinants Question - None

Question

If $\Delta = \begin{vmatrix} \sin x & \sin(x+h) & \sin(x+2h) \ \sin(x+2h) & \sin x & \sin(x+h) \ \sin(x+h) & \sin(x+2h) & \sin x \end{vmatrix}$, find $\lim_{h \to 0} \frac{\Delta}{h^2}$.

Accepted Answer

Step 1: Let $a = \sin x$, $b = \sin(x+h)$, $c = \sin(x+2h)$

Step 2: The determinant can be written as:

$\Delta = \begin{vmatrix} a & b & c \ c & a & b \ b & c & a \end{vmatrix}$

Step 3: For a circulant determinant of this form:

$\Delta = (a+b+c)[(a-b)^2 + (b-c)^2 + (c-a)^2]/2$

Step 4: Using Taylor expansion as $h o 0$:

$a - b = \sin x - \sin(x+h) = -2\cos(x + h/2)\sin(h/2)$

Step 5: Each difference term is $O(h)$, so the numerator is $O(h^2)$ and $\lim_{h o 0} \frac{\Delta}{h^2} = 0$

Question:

Step-by-Step Solution

Master Matrices & Determinants with Mathbee