If A, B are square matrices of same order and B is a skew-symmetric ma

1.	If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A’ BA is skew symmetric.
Answer» A matrix is said to be skew-symmetric if A = -A’ Given, B is a skew-symmetric matrix. ∴ B = -B’ Let C = A’ BA …(1) We have to prove C is skew-symmetric. To prove: C = -C’ As C’ = (A’BA)’ We know that: (AB)’ = B’A’ ⇒ C’ = (A’BA)’ = A’B’(A’)’ ⇒ C’ = A’B’A {∵ (A’)’ = A} ⇒ C’ = A’(-B)A ⇒ C’ = -A’BA …(2) From equation 1 and 2: We have, C’ = -C Thus we say that C = A’ BA is a skew-symmetric matrix.

If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A’ BA is skew symmetric.

Answer»

A matrix is said to be skew-symmetric if A = -A’

Given, B is a skew-symmetric matrix.

∴ B = -B’

Let C = A’ BA …(1)

We have to prove C is skew-symmetric.

To prove: C = -C’

As C’ = (A’BA)’

We know that: (AB)’ = B’A’

⇒ C’ = (A’BA)’ = A’B’(A’)’

⇒ C’ = A’B’A {∵ (A’)’ = A}

⇒ C’ = A’(-B)A

⇒ C’ = -A’BA …(2)

From equation 1 and 2:

We have,

C’ = -C

Thus we say that C = A’ BA is a skew-symmetric matrix.

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If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A’ BA is skew symmetric.

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