If A and B are non-singular square matrices of even order such that AT=A,BT=-B and AB+BA=O (where O is a null matrix ), then

Question

If A and B are non-singular square matrices of even order such that AT=A,BT=-B and AB+BA=O (where O is a null matrix ), then (A) (A B) is skew symmetric matrix (B) A- 1 and B- 1 commute each other (C) (A- 1 B) is symmetric matrix (D) B and A- 1 commute each other

Tardigrade Admin · Accepted Answer

AB=-BA⇒ A- 1AB=-A- 1BA ⇒ B=-A- 1BA⇒ BA- 1=-A- 1B ( A- 1 B)T=B⊤(A⊤)- 1=-BA- 1=A- 1B

Q. If $A$ and $B$ are non-singular square matrices of even order such that $A^{T} = A, B^{T} = - B$ and $A B + B A = O$
(where $O$ is a null matrix ), then

$(A B)$ is skew symmetric matrix

$A^{- 1}$ and $B^{- 1}$ commute each other

$(A^{- 1} B)$ is symmetric matrix

$B$ and $A^{- 1}$ commute each other

$A B = - B A \Rightarrow A^{- 1} A B = - A^{- 1} B A$
$\Rightarrow B = - A^{- 1} B A \Rightarrow B A^{- 1} = - A^{- 1} B$
$(A^{- 1} B)^{T} = B^{⊤} (A^{⊤})^{- 1} = - B A^{- 1} = A^{- 1} B$

Q. If A and B are non-singular square matrices of even order such that AT=A,BT=−B and AB+BA=O (where O is a null matrix ), then

(AB) is skew symmetric matrix

A−1 and B−1 commute each other

(A−1B) is symmetric matrix

B and A−1 commute each other