Q.Simplify $\cos\theta \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix} + \sin\theta \begin{bmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{bmatrix}$.

Concept: Matrix Addition — add corresponding entries after scalar multiplication.

Step 1: Multiply each matrix by its scalar.

First term:

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

Second term:

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

Step 2: Add the two matrices entry-wise.

Top-left: $\cos^2\theta + \sin^2\theta = 1$ …

We have two matrices, each multiplied by a trigonometric scalar. The key is to treat each matrix as a single object: scalar multiplication means every entry inside the matrix gets multiplied by that scalar. Then we add the two resulting matrices entry by entry.

Let’s do it step by step.

1. First term: $\cos\theta \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$ Multiply every entry by $\cos\theta$:

2. Second term: $\sin\theta \begin{bmatrix} \sin\theta & -\cos\theta \\ \cos\theta & \sin\theta \end{bmatrix}$ Multiply every entry by $\sin\theta$:

3. Add the two matrices: Add corresponding entries.

   • Top-left: $\cos^2\theta + \sin^2\theta = 1$
   • Top-right: $\cos\theta \sin\theta + (-\sin\theta \cos\theta) = 0$
   • Bottom-left: $-\cos\theta \sin\theta + \sin\theta \cos\theta = 0$
   • Bottom-right: $\cos^2\theta + \sin^2\theta = 1$

   So the sum is:

   $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$ …

Method: Simplifying a Scalar-Weighted Sum of Matrices With Trig Entries

Use this when scalars (often trig functions) multiply matrices that are then added, and the result simplifies via an identity.

Steps

Step 1: Distribute each scalar into its matrix.

Multiply every entry of each matrix by its scalar: e.g. $\cos\theta$ times the first matrix scales all four entries by $\cos\theta$, giving terms like $\cos^2\theta$ and $\cos\theta\sin\theta$.

Step 2: Add the two matrices entrywise. …

Common Mistakes

Mistake 1: Scaling only part of a matrix by its scalar.

Why it's wrong: the scalar multiplies every entry, so all four positions pick up a factor. Correct approach: distribute the scalar fully before adding.

Mistake 2: Sign errors in the off-diagonal cancellation.

Why it's wrong: the top-right and bottom-left combine as $\cos\theta\sin\theta-\cos\theta\sin\theta=0$; a wrong sign leaves a spurious term. Correct approach: keep each entry's sign from the original matrix before adding. …