Question:

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

Updated On: May 11, 2025
  • Symmetric matrix
  • Skew Symmetric matrix
  • A diagonal matrix
  • Zero matrix
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is A

Solution and Explanation

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

A matrix A is symmetric if AT = 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: (AT) × (AT) = (A × A)T = A2T. Here AT = A, thus A2 = A2T, confirming A2 is symmetric.
  • By induction, suppose Ak is symmetric. Then Ak+1 = Ak × A. Therefore: (Ak+1)T = (Ak × A)T = AT × (Ak)T = A × Ak = Ak+1. Hence, Ak+1 is symmetric.

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

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

Was this answer helpful?
0
0