Question
Which of the following is/are correct ?
A.
$$B'AB$$ is symmetric if $$A$$ is symmetric
B.
$$B'AB$$ is skew-symmetric if $$A$$ is symmetric
C.
$$B'AB$$ is symmetric if $$A$$ is skew-symmetric
D.
None of these
Answer :
$$B'AB$$ is symmetric if $$A$$ is symmetric
Solution :
Let $$A$$ be a symmetric matrix.
Then $$A' = A$$
Now, $$\left( {B'AB} \right)' = B'A'\left( {B'} \right)'.\left[ {\because \left( {AB} \right)' = B'A'} \right]$$
$$\eqalign{
& = B'A'B\left[ {\because \left( {B'} \right)' = B} \right] \cr
& = B'AB\left[ {\because A' = A} \right] \cr} $$
⇒ $$B'AB$$ is a symmetric matrix.
Now, let $$A$$ be a skew-symmetric matrix.
Then, $$A' = - A$$
$$\eqalign{
& \therefore \left( {B'AB} \right)' = B'A'\left( {B'} \right)'\left[ {\because \left( {AB} \right)' = B'A'} \right] \cr
& = B'A'B\left[ {\because \left( {B'} \right)' = B} \right] \cr
& = B'\left( { - A} \right)B\left[ {\because A' = - A} \right] \cr
& = - B'AB \cr} $$
∴ $$B'AB$$ is a skew-symmetric matrix.