Question

If we have a matrix as If A\left( \alpha \right) = \left[ {\begin{array}{*{20}{c}} {\cos \alpha }&{\sin \alpha } \\\ { - \sin \alpha }&{\cos \alpha } \end{array}} \right], then the matrix ${A^2}\left( \alpha \right) =$
$1){\text{ }}A\left( {2\alpha } \right)$
$2){\text{ }}A\left( \alpha \right)$
$3){\text{ }}A\left( {3\alpha } \right)$
$4){\text{ }}A\left( {4\alpha } \right)$

Answer 1

We have to find the value of the matrix ${A^2}\left( \alpha \right)$ . We solve this using the concept of operations of matrices . We should have the knowledge of the cases for which the multiplication of two matrices is possible or not . For solving this problem we should also have the knowledge of the various trigonometric identities for the double angle of sine function , cosine function . We should also have the knowledge about the order of a matrix .

Complete step-by-step solution:
Given : A\left( \alpha \right) = \left[ {\begin{array}{*{20}{c}} {\cos \alpha }&{\sin \alpha } \\\ { - \sin \alpha }&{\cos \alpha } \end{array}} \right]
The order of $A\left( \alpha \right)$is $2 \times 2$

Multiplication of matrices
Let A = \left[ {\begin{array}{*{20}{c}} a&b; \end{array}} \right]and B = \left[ {\begin{array}{*{20}{c}} c \\\ d \end{array}} \right]
Then , the product of AB is given as :
$AB = \left[ {a \times c + b \times d} \right]$
For multiplication the number of columns of the first matrix should be equal to the number of rows of the second matrix .
Using multiplication of matrices , we get

{\cos \alpha }&{\sin \alpha } \\\ { - \sin \alpha }&{\cos \alpha } \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} {\cos \alpha }&{\sin \alpha } \\\ { - \sin \alpha }&{\cos \alpha } \end{array}} \right]$$ $${A^2}\left( \alpha \right) = \left[ {\begin{array}{*{20}{c}} {{{\cos }^2}\alpha - {{\sin }^2}\alpha }&{\sin \alpha \cos \alpha + \cos \alpha \sin \alpha } \\\ { - (\sin \alpha \cos \alpha + \cos \alpha \sin \alpha )}&{{{\cos }^2}\alpha - {{\sin }^2}\alpha } \end{array}} \right]$$ We know that , Double angle formula of sin function and cos function is given as : $$\sin 2x = 2\sin x\cos x$$ $$\cos 2x = {\cos ^2}x - {\sin ^2}x$$ Using the double angle formula , we get ${A^2}(\alpha ) = \left( {\begin{array}{*{20}{c}} {cos2\alpha }&{sin2\alpha } \\\ { - sin2\alpha }&{cos2\alpha } \end{array}} \right)$ From above relation , we get ${A^2}(\alpha ) = A(2\alpha )$ Thus , the value of $${A^2}\left( \alpha \right)$$is $$A\left( {2\alpha } \right)$$ **Hence , the correct option is $$\left( 1 \right)$$.** **Note:** The properties of multiplication of matrices : (1) Associative Law : For any three matrices $$A$$ , $$B$$ and $$C$$ . We have$$\left( {AB} \right)C = A\left( {BC} \right)$$, whenever both sides of the equality are defined . (2) Distributive Law : For any three matrices $$A$$ , $$B$$ and $$C$$. $$\left( i \right)$$ $$A\left( {B + C} \right) = AB + AC$$ , $$(ii)$$ $$\left( {A + B} \right)C = AC + BC$$ whenever both sides of equality are defined . (3) The existence of multiplicative identity : For every square matrix $$A$$, there exists an identity matrix of the same order such that $$IA = AI = A$$.