If A, B are square matrices of the same order and B is a skew-symmetric matrix, then A′BA is:

Concept:

Transpose of a Matrix:

The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix.

For example: \(\rm A=\begin{bmatrix} \rm a & \rm b \\ \rm x & \rm y \end{bmatrix}\Rightarrow A'=\begin{bmatrix} \rm a & \rm x \\ \rm b & \rm y \\ \end{bmatrix}\).

It is denoted by A' or A^T.

Properties of Transpose of a Matrix:

• The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order:

  (AB)' = B'A'

• (ABC)' = C'B'A'

• (A')' = A

Calculation:

It is given that B is a skew-symmetric matrix.

∴ B' = -B

Now, consider the transpose of the product matrix A′BA.

(A′BA)' = A'B'(A')'

= A'(-B)A               [∵ B' = -B]

= -(A'BA)

Since the transpose is equal to its negative, A'BA is a Skew-Symmetric matrix.