Answer
The number of rows in the product of two matrices AB is the number of rows of matrix A and the number of columns in AB is the number of columns of matrix B.
Work Step by Step
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 $2\times 1$.
