Understanding Skew-Symmetric Matrix Properties

The question asks us to identify the type of matrix that results from the inverse of an invertible skew-symmetric matrix (A^-1). Let's break down the properties involved.

What is a Skew-Symmetric Matrix?

A matrix 'A' is called skew-symmetric if its transpose is equal to its negative. Mathematically, this is represented as:

$$ A^T = -A $$

For a matrix A to be skew-symmetric, all its diagonal elements must be zero. For example:

$$ A = \begin{pmatrix} 0 & a & b \\ -a & 0 & c \\ -b & -c & 0 \end{pmatrix} $$

When you transpose this matrix, you swap rows and columns. Notice how each element $A_{ij}$ becomes $A_{ji}$, and because it's skew-symmetric, $A_{ji} = -A_{ij}$. This includes the diagonal elements where $A_{ii} = -A_{ii}$, which implies $2A_{ii} = 0$, so $A_{ii} = 0$.

What is an Invertible Matrix?

A square matrix is invertible if it has a multiplicative inverse, denoted as A^-1. This means that when you multiply the matrix by its inverse, you get the identity matrix (I):

$$ A A^{-1} = A^{-1} A = I $$

An important condition for invertibility is that the determinant of the matrix must not be zero (det(A) ≠ 0).

Deriving the Nature of A^-1

We are given that A is an invertible skew-symmetric matrix. We need to find the property of A^-1.

1. Start with the definition of a skew-symmetric matrix:

   $$ A^T = -A $$

2. Consider the transpose of the inverse matrix, $(A^{-1})^T$. We know a property relating the transpose and inverse of a matrix:

   $$ (A^{-1})^T = (A^T)^{-1} $$

3. Now substitute the skew-symmetric property ($A^T = -A$) into the equation from step 2:

   $$ (A^T)^{-1} = (-A)^{-1} $$

4. We also know another property related to the inverse of a scalar multiple of a matrix: $(-A)^{-1} = -(A^{-1})$.

   Proof for this property: Consider the product $(-A) \times -(A^{-1}) = (-1)A \times (-1)A^{-1} = (-1)(-1) \times A A^{-1} = 1 \times I = I$. Since multiplying $-A$ by $-(A^{-1})$ gives the identity matrix, $-(A^{-1})$ is indeed the inverse of $-A$.

5. Combine the results from step 3 and step 4:

   $$ (A^{-1})^T = -(A^{-1}) $$

This final equation, $(A^{-1})^T = -(A^{-1})$, is the definition of a skew-symmetric matrix. Therefore, if A is an invertible skew-symmetric matrix, its inverse, A^-1, must also be skew-symmetric.

Conclusion

Based on the derivation, the inverse of an invertible skew-symmetric matrix is a skew-symmetric matrix.