Question

If $A$ and $B$ are two matrices such that $A$ has identical rows and $AB$ is defined, then $AB$ has

• A. no identical rows
• B. identical rows
• C. no identical columns
• D. cannot be determined

Answer 1

The correct option is B

identical rows

$A={\left[\begin{array}{ccccc}a& b& c& \cdots & i\\ a& b& c& \cdots & i\\ \cdot & \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot & ⋮\\ a& b& c& \cdots & i\end{array}\right]}_{m\times n},B={\left[\begin{array}{cccc}k& l& \cdots & x\\ w& v& \cdots & y\\ \cdot & \cdot & \cdot & \cdot \\ \cdot & \cdot & \cdot & \cdot \\ q& r& \cdots & s\end{array}\right]}_{n\times p}$

$AB={\left[\begin{array}{cccc}ak+bw+\cdots +iq& al+bv+\cdots +ir& \cdots \cdots & ax+by+\cdots +is\\ ak+bw+\cdots +iq& al+bv+..+ir& \cdots \cdots & ax+by+\cdots +is\\ \cdot & \cdot & \cdot & \\ \cdot & \cdot & \cdot & \\ ak+bw+\cdots +iq& al+bv+\cdots +ir& \cdots \cdots ..& ax+by+\cdots +is\end{array}\right]}_{m\times p}$
Hence, the rows are identical