For all real values of θ, the matrix. A=( cosθ -sinθ sinθ cosθ ) is (a) an involuto

step 1 So in the given question we have a matrix A which is given as cost theta minus sine theta, sine

For all real values of θ, the matrix. A=( cosθ -sinθ ...

For all real values of $\theta$, the matrix. $A=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$ is
(a) an involutory matrix
(b) an idempotent matrix
(c) a nilpotent matrix of index 2
(d) an invertible matrix

Key Concepts

Rotation Matrix

A rotation matrix is a matrix that performs a rotation in the Euclidean plane. It is defined using trigonometric functions (sine and cosine) and rotates vectors by a specified angle. This concept is fundamental in understanding transformations that preserve lengths and angles.

Orthogonal Matrix

An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors. One of the key properties of orthogonal matrices is that multiplying the matrix by its transpose yields the identity matrix. Rotation matrices are a prime example of orthogonal matrices, which explains their ability to preserve the geometric structure of vectors.

Invertible Matrix

An invertible matrix, also known as a non-singular matrix, is one that has a corresponding inverse such that when the two matrices are multiplied, the result is the identity matrix. This concept is important in linear algebra because it guarantees the existence of a unique solution for systems of linear equations associated with the matrix.

Special Matrix Properties (Involutory, Idempotent, Nilpotent)

These properties describe unique behaviors under repeated multiplication of matrices. An involutory matrix satisfies the condition that when squared, it returns the identity matrix. An idempotent matrix, when squared, reproduces itself. A nilpotent matrix becomes the zero matrix when raised to a certain power. Understanding these properties helps in classifying matrices and predicting their behavior under algebraic operations.

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