Question

A square matrix $A$ is said to be orthogonal if $A^{\prime }A=A{A}^{\prime }=I_n,A^{\prime }$ is transpose of $A.$
If $A$ and $B$ are orthogonal matrices, of the same order, then which one of the following is an orthogonal matrix

Answer 1

$(A+B{)}^{\prime }(A+B)=(A^{\prime }+B^{\prime }\left)\right(A+B)$
$=A^{\prime }A+A^{\prime }B+B^{\prime }A+B^{\prime }B=2I_n+A^{\prime }B+B^{\prime }A$

$(A+iB{)}^{\prime }(A+iB)=(A^{\prime }+i{B}^{\prime }\left)\right(A+iB)$
$=A^{\prime }A+i{A}^{\prime }B+i{B}^{\prime }A-B^{\prime }B=i(A^{\prime }B+B^{\prime }A)$

$\left[i\right(A+B\left){]}^{\prime }\right[i(A+B)]=[i(A^{\prime }+B^{\prime })\left]\right[i(A+B)]$
$=-(2I_n+A^{\prime }B+B^{\prime }A)$

$\left(AB{)}^{\prime }\right(AB)=(B^{\prime }{A}^{\prime }\left)\right(AB)$
$=B^{\prime }\left(A^{\prime }A\right)B=B^{\prime }{I}_{n}B=B^{\prime }B=I_n$

Thus only $AB$ is an orthogonal matrix.