## Precalculus (6th Edition) Blitzer

Two matrices can be added only if they have the same numbers of rows and columns. Consider two matrices of order $3\times 2$ $A=\left[ \begin{matrix} a & d \\ b & e \\ c & f \\ \end{matrix} \right]$ and $B=\left[ \begin{matrix} k & n \\ l & o \\ m & p \\ \end{matrix} \right]$. Then, the addition operation on the matrices is performed as below: \begin{align} & A+B=\left[ \begin{matrix} a & d \\ b & e \\ c & f \\ \end{matrix} \right]+\left[ \begin{matrix} k & n \\ l & o \\ m & p \\ \end{matrix} \right] \\ & =\left[ \begin{matrix} a+k & d+n \\ b+l & e+o \\ c+m & f+p \\ \end{matrix} \right] \end{align} For two matrices to be multiplied, the number of columns of the first matrix should be equal to the number of rows of the second matrix. This means that some matrix C of order $m\times n$ can be multiplied with matrix D of order $n\times p$. Then for assumed matrices, A and B, the addition of which is carried out, multiplication is not possible, since the number of columns of matrix A is not equal to the number of rows of matrix B. Thus, it is not necessary that multiplication of matrices be possible if addition of matrices is possible. Therefore, the statement makes sense.