Skew Symmetric Determinant

One can write a transpose of a matrix by interchanging the rows into columns or vice-versa. For a matrix where the matrices are equal to the negative of its transpose, the matrix formed is the skew-symmetric matrix. However, the determinant of such a matrix is defined as the skew-symmetric determinant. Suppose, if a matrix A is defined and its skew-symmetric transpose is given as -AT, then the determinant of this matrix shall be written as det(-AT). A determinant is any scalar quantity (real or complex) associated with any matrix. In the article, the reader will understand the concepts of such matrices, along with examples. 

Skew-symmetric determinant meaning 

1. The skew-symmetric determinant meaning implies that a determinant is a scalar associated with each matrix, enabling scaling/ characterisation of the matrix and holding physical significance.
2. It can be understood if one tries to understand the concept of determinants in matrices. However, for a skew-symmetric matrix, its determinant can be calculated as per the same procedure for any matrix, and thus the value would be a scalar quantity. Consider a 2 x 2 matrix A written as, 4 7 0  -2 . Then the skew-symmetric matrix of A will be written as the negative transpose, which will be, 

-AT = 4 0 7  -2 .

Now the determinant of this matrix will be written as,

4  0 7  -2 as (4 x -2) – (7 x 0) = -8-0= -8

Therefore, the skew-symmetric determinant of the present given matrix is -8. 

Skew-symmetric determinant properties 

There are many interesting properties for the skew-symmetric determinant. Take a look: 

1. If given the order of a matrix as odd, then the determinant of that matrix shall be zero, for example, given a 3 x 3 matrix implying that the order is 3, which is an odd number. Then the determinant of its skew-symmetric form shall come out to be negative. 
2. If given the order of a matrix as even, then the determinant of that matrix shall be non-zero, for example, given a 2 x 2 matrix implying that the order is 2, which is an even number. Then the skew-symmetric determinant form shall be a non-zero perfect square. 

Consider a matrix A= 0 2 -2  0 , then its transpose is AT= 0 -2 2  0 , and 

Det (AT) = (0 x 0)- (2 x -2) = 0 + 4 = 4, which is a perfect square since the order of the matrix is even. 

Skew-symmetric determinant value 

1. A set of linear equations can be solved using the matrix method. For every linear equation, there is a scalar quantity known as the eigenvalue of that equation. 
2. The eigenvalue can have a positive or negative value or can even be an imaginary number. For a matrix M, the Eigen equation may be written as MX = λX, where λ is the eigenvalue of matrix M, and X is the corresponding eigenvector. 
3. For a skew-symmetric matrix, the value of λ will be either 0 or an imaginary number. The eigenvalue concept is essential for the examination point of view as these come in skew-symmetric determinant hot questions. 

Skew-symmetric determinant theorem

1. Suppose M is a skew-symmetric matrix of the order N x N, then the determinant det (MT)= det (-M) = (-1)N det (M). Therefore, for N= odd, the determinant shall be zero, such matrices all called singular according to Jabobi’s theorem. 

Suppose a skew matrix is defined as M= a -b b a, then the det (M) shall be a perfect square according to the properties. 

1. For N=even, then the det (M) = P(M)2, which implies that the skew-symmetric determinant value for an even matrix shall come out to be a perfect square which is always positive. 

Suppose a skew matrix is defined as M= 0 a b -a 0 k -b -k 0, then the det (M) shall be 0 according to the properties. The skew-symmetric determinant hot questions are essential from an exam point of view. 

Conclusion

When a transpose negative is written, the resulting matrix is a skew-symmetric matrix with a unique set of properties. However, when one finds out the determinant of such a matrix, the value can differ as per the order of the matrix. An odd order matrix has determinant zero, while an even order matrix has a determinant non-zero perfect square. The skew-symmetric determinant is described in the text thoroughly, along with a few solved examples. Upon reading the article, the reader shall understand the concepts deeply, along with a few solved examples for the same. The understanding of these concepts is vital in physics and mathematics.