Understanding Skew Symmetric Matrices

A matrix is called skew symmetric if its transpose equals its negative. For a matrix \( A \), this condition is represented as \( A^T = -A \). This property implies specific relationships between the elements of the matrix.

Applying the Skew Symmetric Condition to Matrix A

We are given the matrix \( A = \begin{bmatrix} x & y \\ z & w \end{bmatrix} \).

First, we find the transpose of A (\( A^T \)) by switching rows and columns:

$$ A^T = \begin{bmatrix} x & z \\ y & w \end{bmatrix} $$

Next, we find the negative of A (\( -A \)) by multiplying every element by -1:

$$ -A = \begin{bmatrix} -x & -y \\ -z & -w \end{bmatrix} $$

For A to be skew symmetric, \( A^T \) must equal \( -A \).

$$ \begin{bmatrix} x & z \\ y & w \end{bmatrix} = \begin{bmatrix} -x & -y \\ -z & -w \end{bmatrix} $$

Deriving Properties from Element Comparison

By comparing the elements in the same positions in both matrices, we get the following equations:

• For the element in row 1, column 1: \( x = -x \)
• For the element in row 1, column 2: \( z = -y \)
• For the element in row 2, column 1: \( y = -z \)
• For the element in row 2, column 2: \( w = -w \)

Solving these equations gives us:

• From \( x = -x \), we have \( 2x = 0 \), which means \( x = 0 \).
• From \( w = -w \), we have \( 2w = 0 \), which means \( w = 0 \).
• The conditions \( z = -y \) and \( y = -z \) are equivalent and state that the off-diagonal elements are negatives of each other.

So, a skew-symmetric matrix of this form requires \( x = 0 \) and \( w = 0 \).

Evaluating the Given Options

Let's examine each option based on the derived conditions (\( x=0 \) and \( w=0 \)):

• Option 1: \( x + w = 0 \)
  Substituting the values \( x=0 \) and \( w=0 \), we get \( 0 + 0 = 0 \). This statement is always true for a skew-symmetric matrix.
• Option 2: \( x + z = 0 \)
  Substituting \( x=0 \), we get \( 0 + z = 0 \), meaning \( z = 0 \). This is only true if \( y \) is also 0, which isn't always the case.
• Option 3: \( x - y = 0 \)
  Substituting \( x=0 \), we get \( 0 - y = 0 \), meaning \( y = 0 \). This is not necessarily true.
• Option 4: \( y - z = 0 \)
  This implies \( y = z \). However, the condition is \( y = -z \). This equality only holds if \( y = z = 0 \), which is a specific case, not a general rule.

Final Conclusion on Skew Symmetric Property

The condition derived directly from the definition of a skew-symmetric matrix, specifically concerning the diagonal elements \( x \) and \( w \), is that they must both be zero. Therefore, the statement \( x + w = 0 \) must be true.