Symmetric And Skew Symmetric Matrices

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.

1. (A)
   Applying transpose properties:
   For symmetric matrices, and .
   Thus, . The statement is false.
2. (B) is symmetric matrix
   Calculate transpose:
   Since and are symmetric: ,
   \text{ and } .
   Thus, , so it is symmetric.
3. (C)
   Transpose of a product: holds true for any matrices. Therefore, this is true.
4. (D)
   For symmetric matrices: and .
   But, in general, , thus the statement is false.
5. (E) is skew symmetric matrix
   Compute the transpose:
   Since and are symmetric: and
   \text{ and } .
   , indicating it is skew symmetric.

1. Definition: A square matrix of order is called skew-symmetric if its transpose equals the negative of the matrix itself, i.e., .
2. Elements Property: This implies that for all elements of the matrix:

3. Conclusion: Consequently, in a skew-symmetric matrix, all diagonal elements must be zero, i.e., where .

To determine the nature of , let’s use the properties of symmetric and skew-symmetric matrices.

Symmetric Matrix Property: A matrix is symmetric if .

Since and are symmetric matrices, we know and .

Now, consider :

Since and , this becomes:

This result implies that is a skew-symmetric matrix, as .

Thus, is skew-symmetric.