Geometrical Matrix

Geometry Level 3

$\begin{bmatrix}{\cos\left(\frac{2\pi}7\right)} & {-\sin\left(\frac{2\pi}7\right)} \\ {\sin\left(\frac{2\pi}7\right)} & {\cos\left(\frac{2\pi}7\right)}\end{bmatrix} ^k = \begin{bmatrix}{1} &{0} \\ {0} & {1}\end{bmatrix}$

Find the least positive integral value of $k$ which satisfy the matrix equation above.

The answer is 7.

3 solutions

Otto Bretscher
Sep 4, 2015

The matrix represents a rotation through the angle of $2\pi/7.$

Can you elaborate on this? Pretend I know nothing but English.

Pi Han Goh - 5 years, 6 months ago

Let me be lazy and just show you a link : https://en.wikipedia.org/wiki/Rotation_matrix

In a 2 x 2 matrix of a linear transformation, the first column is the transform of (1,0) and the second column is the transform of (0,1), so, this is basically just the definition of cos and sin as a parametrization of the unit circle. I hope that makes some sense, Comrade...

Otto Bretscher - 5 years, 6 months ago

Thank you, Comrade Otto!

Pi Han Goh - 5 years, 6 months ago

Info Web
Sep 24, 2020

the given matrix say is A, and let 2pi/7 = x

find A^2, it will be the same matrix but angles now are 2x

find A^3, it will be the same matrix but angles now are 3x

so, continuing the pattern, if k = 7, angles become 7x=(2pi/7)= 2pi = 360 degrees = 0 degrees (for trig funtions)

sin(0) = 0, cos(0) = 1, giving an identity matrix and thus giving the result k=7

Janardhanan Sivaramakrishnan
Sep 6, 2015

I think the question should read "least positive integral value". Else, $k=-7,0$ also satisfy the criterion.

Yeah your'e right

Piyush Uniyal - 5 years, 9 months ago

Geometrical Matrix

The answer is 7.

3 solutions

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