If `A=[{:(cosx,sinx),(-sinx,cosx)]` then find |A|

Answer

To find the determinant of the matrix \( A \) given by

\[ A = \begin{pmatrix} \cos x & \sin x \\ -\sin x & \cos x \end{pmatrix} \]

we will follow these steps:

### Step 1: Write the formula for the determinant of a 2x2 matrix
The determinant of a 2x2 matrix

\[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \]

is calculated using the formula:

\[ |A| = ad - bc \]

### Step 2: Identify the elements of matrix \( A \)
In our case, we have:

- \( a = \cos x \)
- \( b = \sin x \)
- \( c = -\sin x \)
- \( d = \cos x \)

### Step 3: Substitute the elements into the determinant formula
Now, we can substitute these values into the determinant formula:

\[ |A| = (\cos x)(\cos x) - (\sin x)(-\sin x) \]

### Step 4: Simplify the expression
This simplifies to:

\[ |A| = \cos^2 x + \sin^2 x \]

### Step 5: Use the Pythagorean identity
Using the Pythagorean identity, we know that:

\[ \cos^2 x + \sin^2 x = 1 \]

### Final Result
Thus, we find that:

\[ |A| = 1 \]