Courses
Courses for Kids
Free study material
Free LIVE classes
More
LIVE
Join Vedantu’s FREE Mastercalss

If A is a matrix such that \[A = \left[ {\begin{array}{*{20}{c}}
  {\cos \alpha }&{\sin \alpha } \\
  { - \sin \alpha }&{\cos \alpha }
\end{array}} \right]\], then find ${A^2}$.

Answer
VerifiedVerified
363.9k+ 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.
Last updated date: 04th Oct 2023
Total views: 363.9k
Views today: 10.63k