First slide

Transpose of matrix

Question

If $A = (\begin{array}{cc} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{array})$ , then

Moderate

A

A is an orthogonal matrix
B

A is a symmetric matrix
C

A is a skew-symmetric matrix
D

none of these

Solution

We have
$A A^{'} = (\begin{array}{cc} \cos θ & - \sin θ \\ \sin θ & \cos θ \end{array}) (\begin{array}{cc} \cos θ & \sin θ \\ - \sin θ & \cos θ \end{array})$
$= (\begin{array}{cc} \cos^{2} θ + \sin^{2} θ & \cos θ \sin θ - \sin θ \cos θ \\ \cos θ \sin θ - \sin θ \cos θ & \sin^{2} θ + \cos^{2} θ \end{array})$
$= (\begin{array}{ll} 1 0 \\ 0 1 \end{array}) = I_{2}$
Similarly, $A^{'} A = I_{2}$
Thus, A is an orthogonal matrix.

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