If A and B are symmetric matrices of the same order, then (AB^t - BA^t) is

CONCEPT:

Symmetric Matrix:

Any real square matrix A = (a_ij) is said to be symmetric matrix if and only if a_ij = a_ji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = A^t then A is said to be a symmetric matrix.

Skew-symmetric Matrix:

Any real square matrix A = (a_ij) is said to be skew-symmetric matrix if and only if a_ij = - a_ji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = - A^t then A is said to be a skew-symmetric matrix.

Note: If A and B are matrices of same order m × n, then (A ± B)^t = A^t ± B^t and (AB)^t = B^t × A^t.

CALCULATION:

Given: A and B are symmetric matrices of the same order

As we know that, if any real square matrix A = (aij) is said to be symmetric matrix if and only if aij = aji, ∀ i and j or in other words we can say that if A is a real square matrix such that A = At then A is said to be a symmetric matrix.

i.e A = A^t and B = B^t

As we know that, if A and B are matrices of same order m × n, then (A ± B)t = At ± Bt and (AB)t = Bt × At.

⇒ (ABt - BAt)^t = (AB^t)^t - (BA^t)^t

⇒ (ABt - BAt)t = BA^t - AB^t

⇒ (ABt - BAt)t = - (AB^t - BA^t)

So, (ABt - BAt) is a skew symmetric matrix.

Hence, option C is the correct answer.