Question:
If $A$ is $2 \times 3$ matrix and $B$ is a matrix such that $A^{\top} B$ and $B A^{\top}$ both are defined, then what is the order of $B$ ?
Solution:
Order of $A=2 \times 3$
Order of $A^{T}=3 \times 2$
Let order of $B=m \times n$
Given : $A^{T} B$ and $B A^{T}$ are defined
If $A^{T}{ }_{3 \times 2} B_{m \times n}$ exists, then the number of columns in $A^{T}$ must be equal to number of rows in $B$.
$\Rightarrow m=2$
If $B_{m \times n} A^{T} 3 \times 2$ exists, then the number of columns in $B$ must be equal to number of rows in $A^{T}$.
$\Rightarrow n=3$
$\therefore$ Order of $B=2 \times 3$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.