Answer
Verified
423.3k+ views
Hint: Multiply the matrix A with itself using the multiplication rule of matrices.
Complete step-by-step answer:
The given matrix is \[A = \left[ {\begin{array}{*{20}{c}}
{\cos \alpha }&{\sin \alpha } \\
{ - \sin \alpha }&{\cos \alpha }
\end{array}} \right]\]. For finding ${A^2}$, we will multiply A with itself. So, we’ll get:
$
\Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
{\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], \\
\Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
{{{\cos }^2}\alpha - {{\sin }^2}\alpha }&{\cos \alpha \sin \alpha + \sin \alpha \cos \alpha } \\
{ - \sin \alpha \cos \alpha + \cos \alpha \left( { - \sin \alpha } \right)}&{ - {{\sin }^2}\alpha + {{\cos }^2}\alpha }
\end{array}} \right], \\
\Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
{{{\cos }^2}\alpha - {{\sin }^2}\alpha }&{2\sin \alpha \cos \alpha } \\
{ - 2\sin \alpha \cos \alpha }&{{{\cos }^2}\alpha - {{\sin }^2}\alpha }
\end{array}} \right], \\
$
We know that $2\sin \alpha \cos \alpha = \sin 2\alpha {\text{ and }}{\cos ^2}\alpha - {\sin ^2}\alpha = \cos 2\alpha $, substituting these value above, we’ll get:
$ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
{\cos 2\alpha }&{\sin 2\alpha } \\
{ - \sin 2\alpha }&{\cos 2\alpha }
\end{array}} \right]$.
Thus, matrix ${A^2}$ is $\left[ {\begin{array}{*{20}{c}}
{\cos 2\alpha }&{\sin 2\alpha } \\
{ - \sin 2\alpha }&{\cos 2\alpha }
\end{array}} \right]$.
Note: For matrix multiplication to exist, it is necessary that the column of the first matrix must be the same as the row of the second matrix otherwise multiplication will not be defined.
Complete step-by-step answer:
The given matrix is \[A = \left[ {\begin{array}{*{20}{c}}
{\cos \alpha }&{\sin \alpha } \\
{ - \sin \alpha }&{\cos \alpha }
\end{array}} \right]\]. For finding ${A^2}$, we will multiply A with itself. So, we’ll get:
$
\Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
{\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], \\
\Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
{{{\cos }^2}\alpha - {{\sin }^2}\alpha }&{\cos \alpha \sin \alpha + \sin \alpha \cos \alpha } \\
{ - \sin \alpha \cos \alpha + \cos \alpha \left( { - \sin \alpha } \right)}&{ - {{\sin }^2}\alpha + {{\cos }^2}\alpha }
\end{array}} \right], \\
\Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
{{{\cos }^2}\alpha - {{\sin }^2}\alpha }&{2\sin \alpha \cos \alpha } \\
{ - 2\sin \alpha \cos \alpha }&{{{\cos }^2}\alpha - {{\sin }^2}\alpha }
\end{array}} \right], \\
$
We know that $2\sin \alpha \cos \alpha = \sin 2\alpha {\text{ and }}{\cos ^2}\alpha - {\sin ^2}\alpha = \cos 2\alpha $, substituting these value above, we’ll get:
$ \Rightarrow {A^2} = \left[ {\begin{array}{*{20}{c}}
{\cos 2\alpha }&{\sin 2\alpha } \\
{ - \sin 2\alpha }&{\cos 2\alpha }
\end{array}} \right]$.
Thus, matrix ${A^2}$ is $\left[ {\begin{array}{*{20}{c}}
{\cos 2\alpha }&{\sin 2\alpha } \\
{ - \sin 2\alpha }&{\cos 2\alpha }
\end{array}} \right]$.
Note: For matrix multiplication to exist, it is necessary that the column of the first matrix must be the same as the row of the second matrix otherwise multiplication will not be defined.
Recently Updated Pages
Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Trending doubts
Select the word that is correctly spelled a Twelveth class 10 english CBSE
What is the z value for a 90 95 and 99 percent confidence class 11 maths CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
What is the past participle of wear Is it worn or class 10 english CBSE
Fill the blanks with proper collective nouns 1 A of class 10 english CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE