Question:

If $A = [\begin{array}{cc} \cos α & \sin α \\ - \sin α & \cos α \end{array}]$ then $A A^{T}$ is equal to: (where $A^{T}$ is the transpose of $A)$

Updated On: Jun 23, 2024

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The Correct Option is B

Explanation:
Given,

\begin{array}{r} A = [\begin{array}{cc} \cos α & \sin α \\ - \sin α & \cos α \end{array}] \end{array}

The transpose of matrix

A

is given by,

\begin{array}{r} A^{T} = [\begin{array}{cc} \cos α & - \sin α \\ \sin α & \cos α \end{array}] \end{array}

Now,

\begin{aligned} A A^{T} = [\begin{array}{cc} \cos α & \sin α \\ - \sin α & \cos α \end{array}] \times [\begin{array}{cc} \cos α & - \sin α \\ \sin α & \cos α \end{array}] \\ \Rightarrow A A^{T} = [\begin{array}{cc} \cos^{2} α + \sin^{2} α & - \cos α \sin α + \cos α \sin α \\ - \cos α \sin α + \cos α \sin α & \sin^{2} α + \cos^{2} α \end{array}] \\ \Rightarrow A A^{T} = [\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}] \end{aligned}

\Rightarrow A A^{T} = I

So,

A A^{T}

is a identify matrix.Hence, the correct option is (B).

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