Skew Symmetric Matrix:
A skew symmetric matrix is a square matrix in which the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal.

Singular Matrix:
A matrix is said to be singular if its determinant is equal to zero. In other words, a matrix is said to be singular if it does not have an inverse.

Odd Order Skew Symmetric Matrix:
An odd order skew symmetric matrix is a skew symmetric matrix of odd dimensions. Let's consider an odd order skew symmetric matrix A of size n x n.

Determinant of Odd Order Skew Symmetric Matrix:
To determine the determinant of an odd order skew symmetric matrix, we can use the fact that the determinant of a skew symmetric matrix is always zero.

Proof:
Let A be an odd order skew symmetric matrix. We can represent A as follows:
A =
[ 0 a b c d ]
[ -a 0 e f g ]
[ -b -e 0 h i ]
[ -c -f -h 0 j ]
[ -d -g -i -j 0 ]

Taking the determinant of A, we can expand it along the first column:
det(A) = 0 * det(M1) - a * det(M2) + b * det(M3) - c * det(M4) + d * det(M5)

Where M1, M2, M3, M4, M5 are the (n-1) x (n-1) submatrices obtained by removing the first row and column of A.

Since A is skew symmetric, we have a = 0, b = 0, c = 0, and d = 0. Therefore, all the terms in the determinant expression become zero.

Hence, det(A) = 0, which implies that the odd order skew symmetric matrix is singular.

Even Order Skew Symmetric Matrix:
An even order skew symmetric matrix is a skew symmetric matrix of even dimensions. Let's consider an even order skew symmetric matrix A of size n x n.

Determinant of Even Order Skew Symmetric Matrix:
To determine the determinant of an even order skew symmetric matrix, we can use the fact that the determinant of a skew symmetric matrix is always zero.

Proof:
Let A be an even order skew symmetric matrix. We can represent A as follows:
A =
[ 0 a b c d ]
[ -a 0 e f g ]
[ -b -e 0 h i ]
[ -c -f -h 0 j ]
[ -d -g -i -j 0 ]
[ . . . . . ]
[ . . . . . ]
[ . . . . . ]

Taking the determinant of A, we can expand it along the first column:
det(A) = 0 * det(M1) - a * det(M2) + b * det(M3) - c * det(M4) + d * det(M5) - ... + (-1)^(n+1) * d * det(Mn)