Question

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

Answer 1

Hint: This question is from the chapter, named Matrix. To get the result, multiply the matrix A by itself. After that apply some trigonometric formulas to reach the desired result.

Complete step by step Solution:
Basically, a matrix is a rectangular array that contains the elements of an object. In other words, we can say that a matrix is a collection of the elements or properties of the objects in the form of a rectangular array. All the elements of an object are arranged in the form of rows and columns.
If the number of rows is equal to the number of the columns, then the matrix is said to be a square matrix. Now,
According to the given question, matrix A is a square matrix it men
Now, we have given that
   \[ \Rightarrow \begin{array}{*{20}{c}}
  {A(\alpha )}& = &{\left[ {\begin{array}{*{20}{c}}
  {\cos \alpha }&{\sin \alpha } \\
  { - \sin \alpha }&{\cos \alpha }
\end{array}} \right]}
\end{array}\]
And we have to find the value of \[{A^2}(a)\]. Therefore, for that purpose, we will,
\[ \Rightarrow \begin{array}{*{20}{c}}
  {{A^2}(a)}& = &{\left[ {\begin{array}{*{20}{c}}
  {\cos \alpha }&{\sin \alpha } \\
  { - \sin \alpha }&{\cos \alpha }
\end{array}} \right]}
\end{array}\left[ {\begin{array}{*{20}{c}}
  {\cos \alpha }&{\sin \alpha } \\
  { - \sin \alpha }&{\cos \alpha }
\end{array}} \right]\]
Now, we will get.
\[ \Rightarrow \begin{array}{*{20}{c}}
  {{A^2}(a)}& = &{\left[ {\begin{array}{*{20}{c}}
  {{{\cos }^2}\alpha - {{\sin }^2}\alpha }&{2\sin \alpha \cos \alpha } \\
  { - 2\sin \alpha \cos \alpha }&{{{\cos }^2}\alpha - {{\sin }^2}\alpha }
\end{array}} \right]}
\end{array}\]
Now we know that
\[ \Rightarrow \begin{array}{*{20}{c}}
  {\cos 2\alpha }& = &{{{\cos }^2}\alpha - {{\sin }^2}\alpha }
\end{array}\] and \[\begin{array}{*{20}{c}}
  {\sin 2\alpha }& = &{2\sin \alpha \cos \alpha }
\end{array}\]
Therefore, from the above matrix, we will get that,
\[ \Rightarrow \begin{array}{*{20}{c}}
  {{A^2}(a)}& = &{\left[ {\begin{array}{*{20}{c}}
  {\cos 2a}&{\sin 2a} \\
  { - \sin 2\alpha }&{\cos 2a}
\end{array}} \right]}
\end{array}\]
Therefore, we can write above matrix as,
\[ \Rightarrow \begin{array}{*{20}{c}}
  {{A^2}(a)}& = &{A(2\alpha )}
\end{array}\]
Now the final answer is \[A(2\alpha )\].

Hence, the correct option is 1.

Note: A matrix is the collection of the elements of the objects in the form of a rectangular array. When we find the multiplication of a matrix, one thing has to be kept in mind that the multiplication of a matrix is possible only when the number of columns of the first matrix is equal to the number of the rows of the second matrix.