**Question:**

If $A, B$ and $C$ are $m \times n, n \times p$ and $p \times q$ matrices respectively such that $(B C) A$ is defined, then $m=$_________

**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 $m \times q$.

The order of matrices $A, B$ and $C$ are $m \times n, n \times p$ and $p \times q$, respectively.

Now,

The order of matrix $B C$ is $n \times q$.

For $(B C) A$ to be defined,

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

If $A, B$ and $C$ are $m \times n, n \times p$ and $p \times q$ matrices respectively such that $(B C) A$ is defined, then $m=$ $q$