Question

If A is a symmetric matrix and n ∈ N, then Aⁿ is :

• A. Symmetric matrix
• B. Skew Symmetric matrix
• C. A diagonal matrix
• D. Zero matrix

Answer 1

The Correct Option is A

Given that A is a symmetric matrix and n ∈ N, we want to determine the nature of Aⁿ.

A matrix A is symmetric if A^T = A, where ^T denotes the transpose of the matrix.

To solve this, consider the matrix power Aⁿ. We need to check if this power remains symmetric.

We begin by establishing a property:

If A is symmetric, then:

• A² = A × A. Since A is symmetric: (A^T) × (A^T) = (A × A)^T = A^2T. Here A^T = A, thus A² = A^2T, confirming A² is symmetric.
• By induction, suppose A^k is symmetric. Then A^k+1 = A^k × A. Therefore: (A^k+1)^T = (A^k × A)^T = A^T × (A^k)^T = A × A^k = A^k+1. Hence, A^k+1 is symmetric.

By induction, Aⁿ is symmetric for any natural number n.

Thus, the correct option is that Aⁿ is a symmetric matrix.