Which of the following is true?

(A) For any orthogonal matrix A, we have
A′A = I
Let B be a matrix such that AB = I.
Now we have
A′ = A′ I                [by property of unit matrix]
= A′(AB) = (A′A)B = IB = B
Therefore, (A′)′(A) = AA′ = AB = I
⇒ A′ is orthogonal.
(B) For any orthogonal matrix A, we have
A′A = I
⇒ |A′A| = |I| ⇒ |A′| |A| = 1
⇒ |A| ≠ 0, i.e., A is non-singular.
(C) Let A and B be two orthogonal matrices, therefore,
(AB)′ (AB) = B′A′AB             [by property of transpose]
= B′(A′A)B                           [by associative law]
= B′(IB) = B′B = I
⇒ AB is orthogonal.
(D) Let A be orthogonal matrix and B be its inverse matrix.
Then, we have,
A′A = I                                                 (1)
and, AB = I = BA                                 (2)
Now, we have,
(AB)′ (AB) = B′A′AB
= B′(A′A)B = B′(IB)B = B′B                    (3)
Also, from equation (2), we have
(AB)′ = I′ = I
i.e., B′B = I                                             [Using equation (3)]
⇒ B is orthogonal.
The correct option is (A), (B), (C) and (D)