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

Answer :

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

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

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

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

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

• which is a unit matrix of order 2 x 2.

Answer 2

Answer :-✨....

To simplify the expression, let's expand the product and combine like terms:

First term:

cos(theta) * [cos(theta) sin(theta)

-sin(theta) cos(theta)]

= [cos^2(theta) sin(theta) cos(theta) - cos(theta) sin(theta) cos^2(theta)

-cos(theta) sin(theta) cos(theta) + sin^2(theta) cos(theta)]

= [cos^2(theta) sin(theta) - cos(theta) sin^2(theta)

-cos(theta) sin(theta) + sin^2(theta) cos(theta)]

= [sin(theta) cos^2(theta) - sin^2(theta) cos(theta)

-cos(theta) sin(theta) + sin^2(theta) cos(theta)]

= [sin(theta) (cos^2(theta) - sin^2(theta))

cos(theta) (sin^2(theta) - sin(theta))]

= [sin(theta) cos(2theta)

cos(theta) sin(2theta)]

Second term:

sin(theta) * [sin(theta) -cos(theta)

cos(theta) sin(theta)]

= [sin^2(theta) - cos^2(theta)

cos(theta) sin(theta) - sin(theta) cos(theta)]

= [-(cos^2(theta) - sin^2(theta))

sin(theta) (cos(theta) - sin(theta))]

= [-cos(2theta)

sin(theta) (cos(theta) - sin(theta))]

Adding the two terms together:

[sin(theta) cos(2theta) - cos(2theta)

cos(theta) sin(2theta) + sin(theta) (cos(theta) - sin(theta))]

= [-cos(2theta)

sin(theta) (cos(theta) - sin(theta))]

Therefore, the simplified expression is:

[-cos(2theta)

sin(theta) (cos(theta) - sin(theta))]

thank you ✨....