Question

Simplify the given sum of matrices: $\cos \theta \left[ \begin{matrix}
   \cos \theta & \sin \theta \\
   -\sin \theta & \cos \theta \\
\end{matrix} \right]+\sin \theta \left[ \begin{matrix}
   \sin \theta & -\cos \theta \\
   \cos \theta & \sin \theta \\
\end{matrix} \right]$.

Answer 1

Hint: We start solving the problem by assigning a variable matrix for the given sum of matrices. We do multiplication wherever required in both of the matrices. We make use of the sum of the matrices to get all in a single matrix. We make use of trigonometric identities to get the final result.

Complete step by step answer:
Given that we need to find the value of sum of the matrices $\cos \theta \left[ \begin{matrix}
   \cos \theta & \sin \theta \\
   -\sin \theta & \cos \theta \\
\end{matrix} \right]+\sin \theta \left[ \begin{matrix}
   \sin \theta & -\cos \theta \\
   \cos \theta & \sin \theta \\
\end{matrix} \right]$. Let us assume the sum is ‘A’.
We have got the value of $A=\cos \theta \left[ \begin{matrix}
   \cos \theta & \sin \theta \\
   -\sin \theta & \cos \theta \\
\end{matrix} \right]+\sin \theta \left[ \begin{matrix}
   \sin \theta & -\cos \theta \\
   \cos \theta & \sin \theta \\
\end{matrix} \right]$ ---(1).
We know that when a variable or function is multiplied to matrix, it multiplies with each and every element of the matrix i.e., $x\times \left[ \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right]=\left[ \begin{matrix}
   x\times a & x\times b \\
   x\times c & x\times d \\
\end{matrix} \right]$.
We have got the value of $A=\left[ \begin{matrix}
   \cos \theta \times \cos \theta & \sin \theta \times \cos \theta \\
   -\sin \theta \times \cos \theta & \cos \theta \times \cos \theta \\
\end{matrix} \right]+\left[ \begin{matrix}
   \sin \theta \times \sin \theta & -\cos \theta \times \sin \theta \\
   \cos \theta \times \sin \theta & \sin \theta \times \sin \theta \\
\end{matrix} \right]$.
We have got the value of $A=\left[ \begin{matrix}
   {{\cos }^{2}}\theta & \sin \theta .\cos \theta \\
   -\sin \theta .\cos \theta & {{\cos }^{2}}\theta \\
\end{matrix} \right]+\left[ \begin{matrix}
   {{\sin }^{2}}\theta & -\cos \theta .\sin \theta \\
   \cos \theta .\sin \theta & {{\sin }^{2}}\theta \\
\end{matrix} \right]$ ---(2).
We know that the sum of two matrices is defined as the sum of corresponding elements from each matrix i.e., $\left[ \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right]+\left[ \begin{matrix}
   p & q \\
   r & s \\
\end{matrix} \right]=\left[ \begin{matrix}
   a+p & b+q \\
   c+r & d+s \\
\end{matrix} \right]$. We use this result in equation (2).
We have got the value of $A=\left[ \begin{matrix}
   {{\cos }^{2}}\theta +{{\sin }^{2}}\theta & \sin \theta .\cos \theta -\cos \theta .\sin \theta \\
   -\sin \theta .\cos \theta +\cos \theta .\sin \theta & {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \\
\end{matrix} \right]$ ---(3).
We know that ${{\cos }^{2}}A+{{\sin }^{2}}A=1$ and $\sin A\cos A=\cos A\sin A$. We use these results in equation (3).
We have got the value of $A=\left[ \begin{matrix}
   1 & \sin \theta .\cos \theta -\sin \theta .\cos \theta \\
   -\sin \theta .\cos \theta +\sin \theta .\cos \theta & 1 \\
\end{matrix} \right]$.
We have got the value of $A=\left[ \begin{matrix}
   1 & 0 \\
   0 & 1 \\
\end{matrix} \right]$.
We know that in an identity matrix, the elements in the principal diagonal of matrix are one and all the other elements in the matrix are zero.
We have got the value of A = I.
We have found the value of the sum of the matrices $\cos \theta \left[ \begin{matrix}
   \cos \theta & \sin \theta \\
   -\sin \theta & \cos \theta \\
\end{matrix} \right]+\sin \theta \left[ \begin{matrix}
   \sin \theta & -\cos \theta \\
   \cos \theta & \sin \theta \\
\end{matrix} \right]$ as Identity matrix ‘I’.

∴ The value of the sum of the 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 Identity matrix ‘I’.

Note: We should not multiply some elements of the matrix while making multiplication of the matrix. We should not make improper multiplications like multiplying the first matrix with $\sin \theta $ and another with $\cos \theta $. We should not know that the Identity matrix is a special matrix in a scalar matrix which has the value of all principal diagonal as 1.