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Question: A square matrix A is said to be orthogonal \(AA' = A'A = {I_n}\). If A and B are orthogonal matrices...

A square matrix A is said to be orthogonal AA′=A′A=InAA' = A'A = {I_n}. If A and B are orthogonal matrices, of the same size, then which one of the following is an orthogonal matrix:
A) AB
B) A+B
C) A+iB
D) i(A+B)

Explanation

Solution

We need to apply the given definition of orthogonal matrix.Then further some transformations allowed with matrices and using the property that the product of two identity matrices is the identity matrix itself ,we get an answer.

Complete step-by-step answer:
It is given that for orthogonal matrix A, we have
AA′=A′A=In……....(1)AA' = A'A = {I_n}……....(1)
Where In{I_n}the identity matrix of the size/ order n. It is also showing that sequences of products can be interchanged, according to the law of commutativity for multiplication.
Let us take two square matrices A and B. Then AB will be the product of these two.
According to matrix complement properties we can write,
(AB)′=B′A′(AB)' = B'A'
Multiplying both side by AB
(AB)′(AB) =(B′A′)(AB) (AB)'(AB) \\\ = (B'A')(AB)
Upon rearranging,
=B′(AA′)BB'(AA')B
=B′InB………….(using equation (1))B'{I_n}B………….\text{(using equation (1))}
Again rearrangements of the matrices,
=(B′B)In(B'B){I_n}
=InIn..........(using equation (1) for matrix B){I_n}{I_n}..........\text{(using equation (1) for matrix B)}
As, we know that the product of two identity matrices is the identity matrix itself.
So, we have
(B′B)In(B'B){I_n}=In{I_n}.
Thus we get finally,
(AB)′(AB)=In(AB)'(AB) = {I_n}
∴\therefore Based on the given fact about orthogonal matrices, AB will be the orthogonal matrix.

So, the correct answer is “Option A”.

Additional Information: A square matrix with real numbers or elements will be termed as an orthogonal matrix if its transpose is equal to its inverse matrix. In other words, we may say that, when the product of a square matrix and its transpose gives the identity matrix of the same order, then that square matrix will be termed an orthogonal matrix.

Note: Matrix is the systematic arrangement of the numbers. It has its own algebra and rules for manipulation. All four basic operations on the matrices are possible with some applicable theorems. Problems based on matrices can be solved by using suitable theorems and transformations.