**Question:**

If $A$ and $B$ are two matrices of orders $a \times 3$ and $3 \times b$ respectively such that $A B$ exists and is of order $2 \times 4$. Then, $(a, b)=$__________

**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 $m \times q$.

$A$ and $B$ are two matrices of orders $a \times 3$ and $3 \times b$, respectively.

So, $A B$ is a matrix of order $a \times b$.

It is given that, $A B$ exists and its order $2 \times 4$.

$\therefore a=2$ and $b=4$

Thus, the ordered pair $(a, b)$ is $(2,4)$.

If $A$ and $B$ are two matrices of orders $a \times 3$ and $3 \times b$ respectively such that $A B$ exists and is of order $2 \times 4$. Then, $(a, b)=$ $(2,4)$