Matrices & Determinants
Matrix as rotation, powers of matrix
nta_pyq_2023_jan
Grade 12
Question:
If $A = \frac{1}{2}\begin{pmatrix} 1 & \sqrt{3} \\ -\sqrt{3} & 1 \end{pmatrix}$, then:
$A^{30} - A^{25} = 2I$
$A^{30} + A^{25} + A = I$
$A^{30} + A^{25} - A = I$
$A^{30} = A^{25}$
Step-by-Step Solution
Key Concept: Recognize A as a rotation matrix with angle $-60°$ (i.e., $A = $ rotation by $-\pi/3$), so $A^n$ is rotation by $-n\pi/3$; $A^6 = I$
A is a rotation by $-60° = -\pi/3$. So $A^6 = I$. $A^{30} = (A^6)^5 = I$. $A^{25} = A^{24}\cdot A = A$. $A^{30} + A^{25} - A = I + A - A = I$. Answer: (3)
Correct Answer: $A^{30} + A^{25} - A = I$