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Mathematics Question on matrix transformation

If A=12[13 −31]A=\frac{1}{2}\begin{bmatrix}1 & \sqrt{3} \\\ -\sqrt{3} & 1\end{bmatrix}, then :

A

A30−A25=2IA ^{30}- A ^{25}=2 I

B

A30=A25A ^{30}= A ^{25}

C

A30+A25−A=IA ^{30}+ A ^{25}- A = I

D

A30+A25+A=IA ^{30}+ A ^{25}+ A = I

Answer

A30+A25−A=IA ^{30}+ A ^{25}- A = I

Explanation

Solution

A=21​[1−3​​3​1​]
A=[cos60∘−sin60∘​sin60∘cos60∘​]
If A=[cosα−sinα​sinαcosα​] Here α=3π​
A2=[cosα−sinα​sinαcosα​][cosα−sinα​sinαcosα​]
=[cos2α−sin2α​sin2αcos2α​]
A30=[cos30α−sin30α​sin30αcos30α​]
A30=[10​01​]=I
A25=[cos25α−sin25α​sin25αcos25α​]=[21​2−3​​​23​​21​​]
A25=A
A25−A=0