Question

Simplify cos theta[ matrix cos theta&sin theta\\ -sin theta&cos theta matrix ]+sin theta[ matrix sin theta&-cos theta\\ cos theta&sin theta matrix ]​

Answer 1

Topic - Matrices

Explanation:

We need to simply,

$\implies\cos \theta\left[ \begin{array}{cc} \cos\theta &\sin \theta \\ \\ -\sin \theta & \cos \theta \end {array}\right] + \sin \theta\left[ \begin{array}{cc} \sin\theta & - \cos \theta \\ \\ \cos \theta & \sin \theta \end {array}\right]$ ‎ ‎ ‎

$\implies \cos \theta\left[ \begin{array}{cc} \cos\theta &\sin \theta \\ \\ -\sin \theta & \cos \theta \end {array}\right] + \sin \theta\left[ \begin{array}{cc} \sin\theta &- \cos \theta \\ \\ \cos \theta &\sin \theta \end {array}\right]$ ‎ ‎ ‎

Now, apply property of multiplication by scaler:-

• $\boxed{\sf k[A_{ij}]_{m\times n}= [k(A_{ij})]_{m\times n} }$ ‎ ‎ ‎

$\implies\left[ \begin{array}{cc} \cos^2\theta &\sin \theta \cdot \cos \theta \\ \\ -\sin \theta \cdot \cos \theta &\cos^2 \theta \end {array}\right] + \left[ \begin{array}{cc} \sin^2\theta &- \cos \theta \cdot \sin \theta\\ \\ \cos \theta \cdot \sin \theta & \sin^2 \theta \end {array}\right]$ ‎ ‎

Now apply property of addition of two matrices according to which, in addition, we add the corresponding elements of matrices. ‎ ‎ ‎

$\implies\left[ \begin{array}{cc} \cos^2\theta + \sin^{2} \theta &\sin \theta \cdot \cos \theta - \cos \theta \cdot \sin \theta \\ \\ -\sin \theta \cdot \cos \theta + \cos \theta \cdot \sin \theta &\cos^2 \theta + \sin^{2} \theta \end {array}\right]$ ‎ ‎ ‎

Now apply trigonometry identity:-

• $\boxed{\sf sin^2\theta + \cos^2\theta = 1}$

$\implies\left[ \begin{array}{cc} 1 &0 \\ \\ 0&1 \end {array}\right]$ ‎ ‎ ‎

Thus resulting an identity matrix of order 2×2. ‎ ‎ ‎

Additional Information :-

Equality of matrices :

Two matrices say A and B are said to be equal iff they have same order and each element of matrix A is equal to matrix B.

$\sf Eg - \begin{bmatrix} 1&3 \\ 2&4\end {bmatrix} = \begin{bmatrix} 1&3 \\ 2&4\end{bmatrix}$

Here, since the order of both the matrices are equal and also each corresponding element of both the matrices are equal, these matrices can be termed as 'Equal'.