If A is a skew-symmetric matrix, then A₂ is a .................

If A is a skew-symmetric matrix, then A² (A squared) is a symmetric matrix. This is an important property of skew-symmetric matrices.

What is a Skew-Symmetric Matrix?

A skew-symmetric matrix is defined as a square matrix A where the transpose of A is equal to the negative of A. In mathematical terms, this means:

• Aᵀ = -A

Properties of A²

When you multiply a skew-symmetric matrix by itself, the resulting matrix A² has the following characteristics:

• A² is symmetric, meaning A²ᵀ = A².
• The diagonal elements of A² are non-negative.

Why is A² Symmetric?

To understand why A² is symmetric, consider the following:

• When you compute A², you are essentially performing A × A.
• Using the property of skew-symmetry, you can show that (A × A)ᵀ = Aᵀ × Aᵀ = (-A) × (-A) = A × A.

Thus, A² retains the symmetry property, confirming that the product of a skew-symmetric matrix with itself results in a symmetric matrix.