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Matrices are rectangular arrays of numbers set up in rows and columns. A 2x2 matrix is one of the simplest forms with two rows and two columns. It looks like this:

• Examples of 2x2 matrices are: - \( \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} \) - \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \)

A 2x2 matrix can represent different transformations in linear algebra, such as rotation or scaling of a plane.
In this context, we deal with the matrix \( \mathbf{A}=\left(\begin{array}{cc} \sin \theta & \cos \theta \ -\cos \theta & \sin \theta \end{array}\right) \), which represents a rotation in the 2D coordinate plane.
The ability to find the inverse of a 2x2 matrix is important because the inverse matrix reverses the effect of the original transformation. This means if the matrix represents a rotation, its inverse will rotate objects back to their original position.

The determinant is a special number that can be calculated from a square matrix. For 2x2 matrices, it helps determine if the matrix has an inverse.

• The formula for the determinant of a 2x2 matrix \( \mathbf{A} = \begin{pmatrix} a & b \ c & d \end{pmatrix} \) is \( \text{det}(\mathbf{A}) = ad - bc \).

In the given problem, our matrix \( \mathbf{A} \) is \( \begin{pmatrix} \sin \theta & \cos \theta \ -\cos \theta & \sin \theta \end{pmatrix} \).
Plug the values into the formula:

• \( a = \sin \theta \)
• \( b = \cos \theta \)
• \( c = -\cos \theta \)
• \( d = \sin \theta \)

The determinant calculates to \( \sin^2 \theta + \cos^2 \theta \), a key identity we'll explore shortly.
If the determinant is not zero, the 2x2 matrix has an inverse, which means the matrix transformation is reversible.

The Pythagorean identity is a fundamental trigonometric identity. It states that for any angle \( \theta \), \( \sin^2 \theta + \cos^2 \theta = 1 \).

• This identity arises from the Pythagorean Theorem in a right-angled triangle, relating the sides of the triangle to the circle's radius.

In the context of matrices:- We calculated the determinant of our matrix as \( \sin^2 \theta + \cos^2 \theta \).
Since the Pythagorean Identity confirms this is 1, the determinant is indeed non-zero.This concept assures us the matrix \( \mathbf{A} \) is invertible, meaning it can effectively reverse its transformation effect when multiplied by its inverse.This identity is crucial in linear algebra and trigonometry, often simplifying complex expressions involving trigonometric functions.

• Remember: inverting a matrix is only possible when the determinant is non-zero. The Pythagorean Identity gives us confidence in certain trigonometric matrices always having non-zero determinants, ensuring they are invertible.