Example, product of two matrices

A numerical example concretely illustrates the row-by-column rule and confirms that the product of matrices is not commutative.

Example — Numerical calculation

$A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix},\quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}.$ $AB = \begin{pmatrix} 1\cdot 5+2\cdot 7 & 1\cdot 6+2\cdot 8 \\ 3\cdot 5+4\cdot 7 & 3\cdot 6+4\cdot 8\end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50\end{pmatrix}.$ $BA = \begin{pmatrix} 5+18 & 10+24 \\ 7+24 & 14+32\end{pmatrix} = \begin{pmatrix} 23 & 34 \\ 31 & 46\end{pmatrix} \ne AB.$

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Topics: Linear systems
Concepts: Matrix product
Skills: Calculate