Q. 485.0( 1 Vote )
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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