Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Question

Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Filo · Accepted Answer

We suppose that A is a symmetric matrix, then A′ =A.........(1)Consider (B′ AB) ′ ={B′ (AB) } ′ =(AB) ′ (B′ ) ′ [(AB) ′ =B′ A′ ] =B′ A′ (B)[(B′ ) ′ =B] =B′ (A′ B) =B′ (AB)  [using (1)]∴(B′ AB) ′ =B′ (AB) Thus, if A is a symmetric matrix, then B′ AB is a symmetric matrix.Now, we suppose that A is a skew-symmetric matrix. Then, A′ =−AConsider(B′ AB) ′ =[B′ (AB) ] ′ =(AB) ′ (B′ ) ′ =(B′ A′ )B=B′ (−A)B=−B′ AB∴(B′ AB) ′ =−B′ ABThus, if A is a skew-symmetric matrix, then B′ AB is a skew-symmetric matrix.Hence, if A is a symmetric or skew-symmetric matrix , then B′ AB is a symmetric or skew-symmetric matrix accordingly.

Question Text	Show that the matrix $B^{'} A B$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric.
Answer Type	Text solution:1
Upvotes	150

Show that the matrix $B^{'} A B$ is symmetric or skew symmetric according as $A$ is symmetric or skew symmetric.

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