Question

Let $A_{2\times m}$ and $B_{n\times 2}$ are two matrices such that $AB$ and $BA$ exist. If $C$ is a matrix such that the order of matrix $BAC$ is $3\times 4,$ then which of the following options is/are true

• A. $m=2,n=3$
• B. $m=n=3$
• C. order of matrix $C$ must be $3\times 4$
• D. matrix addition is not possible between the matrices $A,B$ and $C$

Answer 1

Answer: (B), (C), (D)

matrix addition is not possible between the matrices $A,B$ and $C$

$A_{2\times m},B_{n\times 2}\Rightarrow (AB{)}_{2\times 2}$ possible, if $m=n$
and $(BA{)}_{n\times m}\Rightarrow$ order of $BA$ is $n\times m$
Let order of matrix $C$ is $\alpha \times \beta$
So, $(BA{)}_{n\times m}{C}_{\alpha \times \beta }$ product possible, if $\alpha =m$ and order of product $BAC$ is $n\times \beta$
$\Rightarrow n\times \beta =3\times 4$
$\Rightarrow n=3,\beta =4$
$\therefore n=m=\alpha =3,\beta =4$
$A_{2\times 3},B_{3\times 2}$ and $C_{3\times 4}$ are matrices with different orders. So addition is not possible among $A,B$ and $C$