Understanding Skew-Symmetric Matrices and Their Properties

A matrix \( A \) is defined as skew-symmetric if its transpose \( A^T \) is equal to the negative of the matrix \( -A \). Mathematically, this property is expressed as \( A^T = -A \). This condition implies certain relationships between the elements of the matrix. Specifically, the diagonal elements of a skew-symmetric matrix must be zero, and the element at position \( (i, j) \) must be the negative of the element at position \( (j, i) \) (i.e., \( a_{ij} = -a_{ji} \)).

Analyzing the Given Skew-Symmetric Matrix

We are given the matrix \( A \):

\( A = \begin{bmatrix} 0 & -1 & 3x \\ 1 & y & -5 \\ -6 & 5 & 0 \end{bmatrix} \)

We are told that this matrix is skew-symmetric. Let's verify the properties based on this definition.

Step 1: Find the Transpose of Matrix A

The transpose of a matrix \( A \), denoted as \( A^T \), is obtained by interchanging the rows and columns of \( A \).

\( A^T = \begin{bmatrix} 0 & 1 & -6 \\ -1 & y & 5 \\ 3x & -5 & 0 \end{bmatrix} \)

Step 2: Find the Negative of Matrix A

The negative of a matrix \( A \), denoted as \( -A \), is obtained by multiplying every element of \( A \) by -1.

\( -A = \begin{bmatrix} -(0) & -(-1) & -(3x) \\ -(1) & -(y) & -(-5) \\ -(-6) & -(5) & -(0) \end{bmatrix} = \begin{bmatrix} 0 & 1 & -3x \\ -1 & -y & 5 \\ 6 & -5 & 0 \end{bmatrix} \)

Equating Transpose and Negative Matrix

Since matrix \( A \) is skew-symmetric, we must have \( A^T = -A \). We equate the corresponding elements of the two matrices:

\( \begin{bmatrix} 0 & 1 & -6 \\ -1 & y & 5 \\ 3x & -5 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 1 & -3x \\ -1 & -y & 5 \\ 6 & -5 & 0 \end{bmatrix} \)

Comparing the elements, we get several equations:

• Element (1, 3): \( -6 = -3x \)
• Element (2, 2): \( y = -y \)
• Element (3, 1): \( 3x = 6 \)

Other elements like (1, 2), (2, 1), (2, 3), (3, 2), (1, 1), (3, 3) are already consistent (\( 1=1, -1=-1, 5=5, -5=-5, 0=0, 0=0 \)).

Solving for Variables x and y

From the equations derived by comparing elements:

From \( -6 = -3x \):

\( -3x = -6 \)

\( x = \frac{-6}{-3} \)

\( x = 2 \)

From \( 3x = 6 \):

\( 3x = 6 \)

\( x = \frac{6}{3} \)

\( x = 2 \)

Both equations give the consistent value \( x = 2 \).

From \( y = -y \):

\( y + y = 0 \)

\( 2y = 0 \)

\( y = 0 \)

So, we have found that \( x = 2 \) and \( y = 0 \).

Calculating the Value of \( 5x - y \)

The question asks for the value of the expression \( 5x - y \). We substitute the values of \( x \) and \( y \) we found:

\( 5x - y = 5(2) - 0 \)

\( 5x - y = 10 - 0 \)

\( 5x - y = 10 \)

The value of \( 5x - y \) is 10.

Revision Table: Matrix Properties

Additional Information: Skew-Symmetric Matrix Facts

Here are some additional points about skew-symmetric matrices:

• The main diagonal elements of a skew-symmetric matrix are always zero. This is because for a diagonal element \( a_{ii} \), the condition \( a_{ii} = -a_{ii} \) implies \( 2a_{ii} = 0 \), which means \( a_{ii} = 0 \).
• The determinant of a skew-symmetric matrix of odd order is always zero.
• The determinant of a skew-symmetric matrix of even order is a perfect square.
• Any square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix: \( A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T) \). Here, \( \frac{1}{2}(A + A^T) \) is symmetric and \( \frac{1}{2}(A - A^T) \) is skew-symmetric.