Question

Which of the following is an orthogonal matrix-

A.$\left[\begin{array}{cc}1& 0\\ 1& 0\end{array}\right]$ B. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ C. $\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$ D. $\left[\begin{array}{cc}0& 0\\ 1& 1\end{array}\right]$

Answer 1

Since, a matrix A is said to be orthogonal if $A{A}^T=A^{T}A=I$.
From above the only matrix that satisfies above condition is identity matrix,
Let A=$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $A^T=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$A{A}^T=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$\therefore A{A}^T=I$
Hence, the correct option is B ($\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$)