**Question:**

If $A$ is an $m \times n$ matrix and $B$ is a matrix such that both $A B$ and $B A$ are defined, then the order of $B$ is______

**Solution:**

Let $X=\left[x_{i j}\right]_{m \times n}$ and $Y=\left[y_{i j}\right]_{p \times q}$ be two matrices of order $m \times n$ and $p \times q$. The multiplication of matrices $X$ and $Y$ is defined if number of columns of $X$ is same as the

number of rows of $Y$ i.e. $n=p$. Also, $X Y$ is a matrix of order $m \times q$.

It is given that, $A$ is an $m \times n$ matrix.

Let the order of matrix $B$ be $p \times q$.

For $A B$ to be defined,

$n=p$ ....(1) (Number of columns of $A$ is same as the number of rows of $B$ )

For $B A$ to be defined,

$q=m$ ....(2) (Number of columns of $B$ is same as the number of rows of $A$ )

From (1) and (2), we conclude that the order of matrix $B$ be $n \times m$.

If $A$ is an $m \times n$ matrix and $B$ is a matrix such that both $A B$ and $B A$ are defined, then the order of **$B$ is $n \times m$**