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Multiplication of 2×2 matrices over the real numbers is associative


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 1.4 Exercise 1.4.9

Prove that the binary operation of matrix multiplication of $2 \times 2$ matrices over $\mathbb{R}$ is associative.


Solution: Let $a_i$, $b_i$, $c_i$ be arbitrary real numbers. Then we have \begin{align*}&\ \left[ {a_1 \atop a_3} {a_2 \atop a_4} \right] \cdot \left( \left[ {b_1 \atop b_3} {b_2 \atop b_4} \right] \cdot \left[ {c_1 \atop c_3} {c_2 \atop c_4} \right] \right)\\=&\ \left[ {a_1 \atop a_3} {a_2 \atop a_4} \right] \cdot \left[ {{b_1 c_1 + b_2 c_3} \atop {b_3c_1 + b_4c_3}}\ {{b_1 c_2 + b_2 c_4} \atop {b_3 c_2 + b_4 c_4}} \right]\\ =&\ \left[ {{a_1b_1c_1 + a_1b_2c_3 + a_2b_3c_1 + a_2b_4c_3} \atop {a_3b_1c_1 + a_3b_2c_3 + a_4b_3c_1 + a_4b_4c_3}}\ {{a_1b_1c_2 + a_1b_2c_4 + a_2b_3c_2 + a_2b_4c_4} \atop {a_3b_1c_2 + a_3b_2c_4 + a_4b_3c_2 + a_4b_4c_4}} \right]\\ =&\ \left[ {{a_1b_1 + a_2b_3} \atop {a_3b_1 + a_4b_3}}\ {{a_1b_2 + a_2b_4} \atop {a_3b_2 + a_4b_4}} \right] \cdot \left[ {c_1 \atop c_3} {c_2 \atop c_4} \right]\\=&\ \left (\left[ {a_1 \atop a_3} {a_2 \atop a_4} \right] \cdot \left[ {b_1 \atop b_3} {b_2 \atop b_4} \right] \right) \cdot \left[ {c_1 \atop c_3} {c_2 \atop c_4} \right].\end{align*}So multiplication of $2 \times 2$ matrices over $\mathbb{R}$ is indeed associative.

Linearity

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