Question

A is a matrix of order $3\times 2$ and B is a matrix of order $2\times 3$. If AB = C and BA = D, then

• A. det (C) =0
• B. $det\left(C\right).det\left(D\right)\ne 0$
• C. det (C). det (D) = 0
• D. $det\left(C\right)\ne 0$

Answer 1

Answer: (A), (C)

The correct options are
A det (C) =0
C det (C). det (D) = 0

Let $A=\left[\begin{array}{cc}a& b\\ c& d\\ e& f\end{array}\right]B=\left[\begin{array}{ccc}p& q& r\\ s& t& u\end{array}\right]$
$AB=\left[\begin{array}{ccc}ap+bs& aq+bt& ar+bu\\ cp+ds& cq+dt& cr+du\\ ep+fs& eq+ft& er+fu\end{array}\right]$
$AB=\left|\begin{array}{ccc}a& b& 0\\ c& d& 0\\ e& f& 0\end{array}\right|\times \left|\begin{array}{ccc}p& q& r\\ s& t& 4\\ 0& 0& 0\end{array}\right|$
$|AB|=0\Rightarrow |C|=0$
$BA=D\to$ can take any value.