Precalculus (6th Edition) Blitzer

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: \begin{align} & AB=C \\ & C={{\left[ {{c}_{ik}} \right]}_{m\times k}} \end{align} 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 $X={{\left[ \begin{matrix} k & n \\ l & o \\ m & p \\ \end{matrix} \right]}_{3\times 2}}$ Then, $AX={{\left[ \begin{matrix} a & b \\ c & d \\ \end{matrix} \right]}_{2\times 2}}{{\left[ \begin{matrix} k & n \\ l & o \\ m & p \\ \end{matrix} \right]}_{3\times 2}}$ is not possible because the element a is multiplied with k and b with l, but there is no element left in the first row of matrix A to be multiplied with element m of the matrix X. Similarly, the element c is multiplied with n and d with o, but there is no element left in first row of matrix A to be multiplied with m of the matrix X.