Question

The given multiplication of matrices \[\cos \theta \left[ \begin{matrix}
   \cos \theta & \sin \theta \\
   -\sin \theta & \cos \theta \\
\end{matrix} \right]+\sin \theta \left[ \begin{matrix}
   \sin \theta & -\cos \theta \\
   \cos \theta & \sin \theta \\
\end{matrix} \right]\] is equal to: -
(a) \[\left[ \begin{matrix}
   -1 & 0 \\
   0 & -1 \\
\end{matrix} \right]\]
(b) \[\left[ \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right]\]
(c) \[\left[ \begin{matrix}
   1 & 1 \\
   1 & 1 \\
\end{matrix} \right]\]
(d) \[-\left[ \begin{matrix}
   1 & 1 \\
   1 & 1 \\
\end{matrix} \right]\]

Answer 1

Hint: Assume the two given matrices as matrix A and matrix B. Use the property of multiplication of a scalar with a matrix that if any number is multiplied with a matrix then every element gets multiplied with that number. Multiplied \[\cos \theta \] with A and \[\sin \theta \] with B. Now, apply the addition property of matrices. Add the \[{{a}_{11}}\] element of the first matrix with the corresponding \[{{a}_{11}}\] element of the second matrix and do the same for other elements. Simplify the terms to get the answer. Use the identity: - \[{{\sin }^{2}}x+{{\cos }^{2}}x=1\].

Complete step-by-step solution
We have been provided with the expression: -
\[\Rightarrow \cos \theta \left[ \begin{matrix}
   \cos \theta & \sin \theta \\
   -\sin \theta & \cos \theta \\
\end{matrix} \right]+\sin \theta \left[ \begin{matrix}
   \sin \theta & -\cos \theta \\
   \cos \theta & \sin \theta \\
\end{matrix} \right]\]
Let us assume: - \[\left[ \begin{matrix}
   \cos \theta & \sin \theta \\
   -\sin \theta & \cos \theta \\
\end{matrix} \right]=A\] and \[\left[ \begin{matrix}
   \sin \theta & -\cos \theta \\
   \cos \theta & \sin \theta \\
\end{matrix} \right]=B\].
Now, we have to find \[\cos \theta .A+\sin \theta .B\]. Since \[\sin \theta \] and \[\cos \theta \] are scalars and we know that if any scalar is multiplied to a matric then all the elements of that matrix are multiplied by that scalar. Therefore, we have,
\[\Rightarrow \cos \theta .A=\cos \theta .\left[ \begin{matrix}
   \cos \theta & \sin \theta \\
   -\sin \theta & \cos \theta \\
\end{matrix} \right]\]
\[\Rightarrow \cos \theta .A=\left[ \begin{matrix}
   {{\cos }^{2}}\theta & \sin \theta \cos \theta \\
   -\sin \theta \cos \theta & {{\cos }^{2}}\theta \\
\end{matrix} \right]\] - (1)
And \[\sin \theta .B=\sin \theta .\left[ \begin{matrix}
   \sin \theta & -\cos \theta \\
   \cos \theta & \sin \theta \\
\end{matrix} \right]\]
\[\Rightarrow \sin \theta .B=\left[ \begin{matrix}
   {{\sin }^{2}}\theta & -\sin \theta \cos \theta \\
   \sin \theta \cos \theta & {{\sin }^{2}}\theta \\
\end{matrix} \right]\] - (2)
Adding equations (1) and (2), we get,
\[\Rightarrow \cos \theta .A+\sin \theta .B=\left[ \begin{matrix}
   {{\cos }^{2}}\theta & \sin \theta \cos \theta \\
   -\sin \theta \cos \theta & {{\cos }^{2}}\theta \\
\end{matrix} \right]+\left[ \begin{matrix}
   {{\sin }^{2}}\theta & -\sin \theta \cos \theta \\
   \sin \theta \cos \theta & {{\sin }^{2}}\theta \\
\end{matrix} \right]\]
Now, applying the property of addition of two matrices which says that elements of first matrix should be added to the corresponding elements of second matrix, we get,
\[\Rightarrow \cos \theta .A+\sin \theta .B=\left[ \begin{matrix}
   {{\cos }^{2}}\theta +{{\sin }^{2}}\theta & \sin \theta \cos \theta +\left( -\sin \theta \cos \theta \right) \\
   -\sin \theta \cos \theta +\sin \theta \cos \theta & {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \\
\end{matrix} \right]\]
Cancelling the like terms and using the identity: - \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\], we get,
\[\Rightarrow \cos \theta .A+\sin \theta .B=\left[ \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right]\]
Hence, option (b) is the correct answer.

Note: One may note that we can also solve this question by assigning some particular value to \[\theta \] like \[{{0}^{\circ }},{{90}^{\circ }}\] or \[{{45}^{\circ }}\]. We already know the values of sine and cosine functions of these angles. This process will replace trigonometric functions with their numerical values and we will be able to solve the question more easily. But remember that this process can only be applied if options are provided just like in the above question.