Math, asked by jishanmansoori792, 1 day ago

6. 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 ]

Answers

Answered by SparklingBoy
195

Appropriate Question :

Simply the following :

 \sf cos \theta \begin{bmatrix} \sf cos \theta& \sf sin \theta \\  \\   \sf - sin \theta& \sf cos \theta\end{bmatrix} + sin  \theta \begin{bmatrix} \sf sin \theta& \sf - cos \theta \\  \\   \sf cos \theta& \sf sin \theta\end{bmatrix} \\  \\

Answer :

The resulting matrix is a unit matrix of order 2 x 2.

Step by Step Explanation :

\rm cos \theta \begin{bmatrix} \rm cos \theta& \rm sin \theta \\  \\   \rm - sin \theta& \rm cos \theta\end{bmatrix} + sin  \theta \begin{bmatrix} \rm sin \theta& \rm - cos \theta \\  \\   \rm cos \theta& \rm sin \theta\end{bmatrix} \\  \\

 = \begin{bmatrix} \rm cos {}^{2}  \theta& \rm sin \theta cos \theta\\  \\   \rm - sin \theta cos \theta& \rm cos {}^{2}  \theta\end{bmatrix} + \begin{bmatrix} \rm  {sin}^{2}  \theta& \rm  - sin \theta cos \theta\\  \\   \rm  sin \theta cos \theta& \rm  {sin}^{2}  \theta\end{bmatrix} \\  \\

 = \begin{bmatrix} \rm cos  {}^{2} \theta +  {sin}^{2} \theta & \rm sin \theta cos \theta - sin \theta cos \theta\\  \\   \rm - sin \theta cos \theta +sin \theta cos \theta & \rm  {sin}^{2}  \theta + cos^2 \theta \end{bmatrix} \\  \\

 = \begin{bmatrix} \rm  \: 1 \: & \:  \rm 0   \: \\ \\   \rm0& \rm 1\end{bmatrix} \\  \\

which is a unit matrix of order 2 x 2.

Concepts Used :

  • If any constant or variable is in multiplication with whole matrix then resulting Matrix contains elements of of initial Matrix with multiplied constant or variable.

  • Addition of two Matrix and addition of corresponding elements of two matrix is a same process.
Answered by takename465
19

Step-by-step explanation:

given :

  • 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

to find :

  • matrix sin theta&-cos theta\\ cos theta&sin theta matrix ]

let us do it :

  • please check the attached file please
Attachments:
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