Evaluate :

$\begin{bmatrix$}[r] cos 45° & sin 30° \ \sqrt{2}cos 0° & sin 0° \end{bmatrix}\begin{bmatrix}[r] sin 45° & cos 90° \ sin 90° & cot 45° \end{bmatrix}

Given,

$\Rightarrow \begin{bmatrix$}[r] cos 45° & sin 30° \ \sqrt{2}cos 0° & sin 0° \end{bmatrix}\begin{bmatrix}[r] sin 45° & cos 90° \ sin 90° & cot 45° \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} \ \sqrt{2}(1) & 0 \end{bmatrix}\begin{bmatrix}[r] \dfrac{1}{\sqrt{2}} & 0 \ 1 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] \dfrac{1}{\sqrt{2}} & \dfrac{1}{2} \ \sqrt{2} & 0 \end{bmatrix}\begin{bmatrix}[r] \dfrac{1}{\sqrt{2}} & 0 \ 1 & 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] \dfrac{1}{\sqrt{2}} \times \dfrac{1}{\sqrt{2}} + \dfrac{1}{2} \times 1 & \dfrac{1}{\sqrt{2}} \times 0 + \dfrac{1}{2} \times 1 \ \sqrt{2} \times \dfrac{1}{\sqrt{2}} + 0 \times 1 & \sqrt{2} \times 0 + 0 \times 1 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] \dfrac{1}{2} + \dfrac{1}{2} & 0 + \dfrac{1}{2} \ 1 + 0 & 0 + 0 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 1 & \dfrac{1}{2} \ 1 & 0 \end{bmatrix} \\[1em] \Rightarrow \begin{bmatrix}[r] 1 & 0.5 \ 1 & 0 \end{bmatrix}

Hence, $\begin{bmatrix$}[r] cos 45° & sin 30° \ \sqrt{2}cos 0° & sin 0° \end{bmatrix}\begin{bmatrix}[r] sin 45° & cos 90° \ sin 90° & cot 45° \end{bmatrix} = \begin{bmatrix}[r] 1 & 0.5 \ 1 & 0 \end{bmatrix}.