Basic Technical Mathematics, 11th Edition

16.2 Multiplication of Matrices

• Multiplication of Matrices • Identity Matrix • Inverse of a Matrix

The definition for the multiplication of matrices does not have an intuitive basis. However, through the solution of a system of linear equations we can, at least in part, show why multiplication is defined as it is. Consider Example 1.

EXAMPLE 1 Reasoning for definition of multiplication

If we solve the system of equations

$$\begin{array}{rcrcl}2x& \hspace{0.17em}+\hspace{0.17em}& y& \hspace{0.17em}=\hspace{0.17em}& 1\\ 7x& \hspace{0.17em}+\hspace{0.17em}& 3y& \hspace{0.17em}=\hspace{0.17em}& 5\end{array}$$

we get $x\hspace{0.17em}=\hspace{0.17em}2\hspace{0.17em},\hspace{0.17em}y\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}3.$ Checking this solution in each of the equations, we get

$$\begin{array}{l}2(2)\hspace{0.17em}+\hspace{0.17em}1(\hspace{0.17em}-\hspace{0.17em}3)\hspace{0.17em}=\hspace{0.17em}1\\ 7(2)\hspace{0.17em}+\hspace{0.17em}3(\hspace{0.17em}-\hspace{0.17em}3)\hspace{0.17em}=\hspace{0.17em}5\end{array}$$

Let us represent the coefficients of the equations by the matrix $\hspace{0.17em}[\hspace{0.17em}\begin{array}{ll}2& 1\\ 7& 3\end{array}\hspace{0.17em}]\hspace{0.17em}$ and the solutions by the matrix $\hspace{0.17em}[\hspace{0.17em}\begin{array}{r}2\\ \hspace{0.17em}-\hspace{0.17em}3\end{array}\hspace{0.17em}]\hspace{0.17em}\hspace{0.17em}.\hspace{0.17em}$ If we now indicate the multiplications of these matrices and ...