## Precalculus (6th Edition) Blitzer

The statement does not make sense. The correct statement is, “The $m\times n$ matrix and an $n\times p$ matrix are multiplied by multiplying corresponding elements and then adding them.”
The product of two matrices is defined as below: Consider a matrix A of $m\times n$ order and another matrix B of $n\times p$ order. To find the ${{i}^{th}}$ row and ${{j}^{th}}$ column of the product AB, each element in the ${{i}^{th}}$ row of A is multiplied by the corresponding element in the ${{j}^{th}}$ column of B, and then these products are added. The product matrix AB is an $m\times p$ order matrix, that is, if $A={{\left[ {{a}_{ij}} \right]}_{m\times n}}$ and $B={{\left[ {{b}_{jp}} \right]}_{n\times p}}$, then \begin{align} & AB=C \\ & C={{\left[ {{c}_{ip}} \right]}_{m\times p}} \end{align} Here, ${{c}_{ip}}=\sum\limits_{j=1}^{n}{{{a}_{ij}}{{b}_{jp}}}$ 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 \begin{align} & AX={{\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]}_{2\times 2}}{{\left[ \begin{matrix} x \\ y \\ \end{matrix} \right]}_{2\times 1}} \\ & ={{\left[ \begin{matrix} ax+by \\ cx+dy \\ \end{matrix} \right]}_{2\times 1}} \end{align} The matrix multiplication cannot be performed by simply multiplying the elements. The elements are then added. Therefore, the statement does not make sense. The correct statement is, “The $m\times n$ matrix and an $n\times p$ matrix are multiplied by multiplying corresponding elements and then adding them.”