Question

If A is a skew-symmetric matrix and n is an odd natural number, write whether Aⁿ is symmetric or skew-symmetric or neither of the two.

Answer 1

Since A is given as a skew-symmetric matrix, it satisfies the property:

A^T = −A

Taking the Transpose of Aⁿ:

Using the laws of matrix transposes, the transpose operation can be swapped with the exponent:

(Aⁿ)^T = (A^T)ⁿ

Substituting A^T = −A

(A^T)ⁿ = (−A)ⁿ

Applying the property of the odd power:

Since n is an odd natural number, (−1)n = −1.

Therefore, (A)ⁿ = (−1)ⁿ . Aⁿ

Because (Aⁿ)^T = −Aⁿ, the matrix Aⁿ satisfies the definition of a skew-symmetric matrix.