## Precalculus (6th Edition) Blitzer

The order of a matrix obtained by the product of two matrices is defined as follows: Consider a matrix A of $m\times n$ order and another matrix B of $n\times k$ order. To find the ${{i}^{th}}$ row and ${{j}^{th}}$ column of the product AB, each element in the ${{i}^{th}}$ row of the matrix A is multiplied by the corresponding element in the ${{j}^{th}}$ column of the matrix B, and then obtained products of corresponding elements are added. The product matrix AB is an $m\times k$ order matrix, that is, if $A={{\left[ {{a}_{ij}} \right]}_{m\times n}}$ and $B={{\left[ {{b}_{jk}} \right]}_{n\times k}}$, then: Consider $AB=C$ Therefore, $C={{\left[ {{c}_{ik}} \right]}_{m\times k}}$ The order of matrix C is $\text{number of rows of matrix }A\times \text{number of columns of matrix }B$ Here, ${{c}_{ik}}=\sum\limits_{j=1}^{n}{{{a}_{ij}}{{b}_{jk}}}$ Let $A={{\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]}_{2\times 2}}$ and $X={{\left[ \begin{matrix} x \\ y \\ \end{matrix} \right]}_{2\times 1}}$ Then: Calculate AX: \begin{align} & AX=\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]\left[ \begin{matrix} x \\ y \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} ax+by \\ cx+dy \\ \end{matrix} \right] \end{align} In the obtained matrix, the number of rows is two and the number of columns is one. Hence, the order of matrix AX is $2\times 1$.