Simplify : $\cos heta\begin{bmatrix} \cos heta & \sin heta \ -\sin heta & \cos heta \end{bmatrix} + \sin heta\begin{bmatrix} \sin heta & -\cos heta \ \cos heta & \sin heta \end{bmatrix}$

**OR**

Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \ -1 & 2 & 1 \ 5

Question

Simplify : $\cos	heta\begin{bmatrix} \cos	heta & \sin	heta \ -\sin	heta & \cos	heta \end{bmatrix} + \sin	heta\begin{bmatrix} \sin	heta & -\cos	heta \ \cos	heta & \sin	heta \end{bmatrix}$

**OR**

Show that the matrix $A = \begin{bmatrix} 1 & -1 & 5 \ -1 & 2 & 1 \ 5 & 1 & 3 \end{bmatrix}$ is a symmetric matrix.

Accepted Answer

>

> [!TLDR]
> Multiply the scalars into each matrix, add corresponding entries, and simplify using $\sin^2	heta+\cos^2	heta=1$; the result is the identity matrix.
>
> $$\cos	heta\begin{bmatrix} \cos	heta & \sin	heta \ -\sin	heta & \cos	heta \end{bmatrix} + \sin	heta\begin{bmatrix} \sin	heta & -\cos	heta \ \cos	heta & \sin	heta \end{bmatrix}$$
>
> **Step 1: Scale each matrix.**
> $$= \begin{bmatrix} \cos^2	heta & \sin	heta\cos	heta \ -\sin	heta\cos	heta & \cos^2	heta \end{bmatrix} + \begin{bmatrix} \sin^2	heta & -\sin	heta\cos	heta \ \sin	heta\cos	heta & \sin^2	heta \end{bmatrix}$$
>
> **Step 2: Add corresponding entries.**
>
> Entry $(1,1)$: $\cos^2	heta+\sin^2	heta = 1$
>
> Entry $(1,2)$: $\sin	heta\cos	heta - \sin	heta\cos	heta = 0$
>
> Entry $(2,1)$: $-\sin	heta\cos	heta+\sin	heta\cos	heta = 0$
>
> Entry $(2,2)$: $\cos^2	heta+\sin^2	heta = 1$
>
> $$= \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = I$$
> …

Time	Apples	Bananas
Morning	10	5
Afternoon	8	12

Concept understanding — Matrix Addition

Matrix Addition: The Intuition

The Precise Statement

The One Rule You Cannot Break

Properties That Feel Natural

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