An odd order skew symmetric matrix is a square matrix with an odd number of rows and columns where the elements on the main diagonal are all zero, and the elements below the main diagonal are the negative of the corresponding elements above the diagonal.
Here's a breakdown of the key characteristics:
- Odd Order: The matrix has an odd number of rows and columns (e.g., 3x3, 5x5, 7x7, etc.).
- Skew Symmetric: The transpose of the matrix is equal to the negative of the original matrix. This means that if A is the matrix, then A<sup>T</sup> = -A.
- Zero Diagonal: All elements on the main diagonal of the matrix are zero.
Example:
| 0 2 -1 |
| -2 0 3 |
| 1 -3 0 |This is a 3x3 skew symmetric matrix. Notice that:
- The main diagonal has all zeros.
- The element at position (1, 2) is 2, while the element at (2, 1) is -2.
- The element at position (1, 3) is -1, while the element at (3, 1) is 1.
Practical Insights:
- Skew symmetric matrices are frequently used in physics and engineering to represent rotations and angular velocities.
- The determinant of any odd order skew symmetric matrix is always zero.
Solutions:
- You can create an odd order skew symmetric matrix by starting with an odd-sized square matrix with zeros on the main diagonal. Then, fill in the upper triangular part with arbitrary values. The lower triangular part is then filled with the negatives of the corresponding elements above the diagonal.
