ISI2015-DCG-33

Question

Suppose $A$ and $B$ are orthogonal $n \times n$ matrices. Which of the following is also an orthogonal matrix? Assume that $O$ is the null matrix of order $n \times n$ and $I$ is the identity matrix of order $n$.

• A. $AB-BA$
• B. $\begin{pmatrix} A & O \\ O & B \end{pmatrix}$
• C. $\begin{pmatrix} A & I \\ I & B \end{pmatrix}$
• D. $A^2 – B^2$

Answer 1

The product of $2$ orthogonal matrices is an orthogonal matrix and the determinant of an orthogonal matrix $= +1$ or $-1$

 ${\begin{vmatrix} A &O \\ O & B \end{vmatrix}} = |AB|=1$

So this means ${\begin{pmatrix} A &O \\ O & B \end{pmatrix}} $ is also an orthogonal matrix.

Hence $B.$ is correct answer.