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[23−5023−523−5]⋅[−10+225−525−5]\left[\begin{array}{cc}\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}} & 0\\\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}} & \frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}\end{array}\right]\cdot \left[\begin{array}{c}\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\end{array}\right]

The calculator will multiply the 22x22 matrix [23−5023−523−5]\left[\begin{array}{cc}\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}} & 0\\\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}} & \frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}\end{array}\right] by the 22x11 matrix [−10+225−525−5]\left[\begin{array}{c}\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\end{array}\right], with steps shown.

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Your Input

Calculate [23−5023−523−5]⋅[−10+225−525−5].\left[\begin{array}{cc}\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}} & 0\\\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}} & \frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}\end{array}\right]\cdot \left[\begin{array}{c}\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\end{array}\right].

Solution

[23−5023−523−5]⋅[−10+225−525−5]=[(23−5)⋅(−10+225−5)+(0)⋅(25−5)(23−5)⋅(−10+225−5)+(23−5)⋅(25−5)]=[1−525−253−525−25]\left[\begin{array}{cc}{\color{Red}\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}} & {\color{Brown}0}\\{\color{Magenta}\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}} & {\color{Peru}\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}}\end{array}\right]\cdot \left[\begin{array}{c}{\color{Violet}\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}}\\{\color{Crimson}\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}}\end{array}\right] = \left[\begin{array}{c}{\color{Red}\left(\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}\right)}\cdot {\color{Violet}\left(\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}\right)} + {\color{Brown}\left(0\right)}\cdot {\color{Crimson}\left(\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\right)}\\{\color{Magenta}\left(\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}\right)}\cdot {\color{Violet}\left(\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}\right)} + {\color{Peru}\left(\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}\right)}\cdot {\color{Crimson}\left(\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\right)}\end{array}\right] = \left[\begin{array}{c}\frac{1 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}}\\\frac{3 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}}\end{array}\right]

Answer

[23−5023−523−5]⋅[−10+225−525−5]=[1−525−253−525−25]≈[−0.850650808352040.525731112119134]\left[\begin{array}{cc}\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}} & 0\\\frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}} & \frac{\sqrt{2}}{\sqrt{3 - \sqrt{5}}}\end{array}\right]\cdot \left[\begin{array}{c}\frac{- \sqrt{10} + \sqrt{2}}{2 \sqrt{5 - \sqrt{5}}}\\\frac{\sqrt{2}}{\sqrt{5 - \sqrt{5}}}\end{array}\right] = \left[\begin{array}{c}\frac{1 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}}\\\frac{3 - \sqrt{5}}{2 \sqrt{5 - 2 \sqrt{5}}}\end{array}\right]\approx \left[\begin{array}{c}-0.85065080835204\\0.525731112119134\end{array}\right]A