Question

Suppose P and Q are two different matrices of order $3\times n$ and $n\times p$, then the order of the matrix $P\times Q$ is:
$\left(a\right)3\times p$
$\left(b\right)p\times 3$
$\left(c\right)n\times n$
$\left(d\right)3\times 3$

Answer 1

For solving the questions related to matrices it is important for us to know several properties of it. We should also know some results of matrix multiplication. Also, see that we do not need to perform the whole multiplication of the two matrices but only the order is to be found. For multiplication to be possible for two matrices the number of columns of first matrix should be equal to the number of rows of the second one and the resultant matrix obtained has number of rows as the number of rows of the first one and number of columns as the number of columns of the second one.

Complete step-by-step solution:
We know that the matrix multiplication happens in a certain way. To produce the first column first row entry of the resultant matrix, we need to multiply the first row of the first matrix and first column of the second matrix element-wise and add the quantities obtained. This is how the multiplication is done. It is important to see that the number of columns of the first matrix should be equal to the number of rows of the second matrix. So if we have a matrix of order $m\times n$ and we have to multiply it with a matrix of order $n\times p$ then the resultant matrix will be of order$m\times p$.
Here, we are given the order $3\times n$ and $n\times p$, so the matrix produced after multiplying these matrices will have the order $3\times p$. Hence, option $\left(a\right)$ is correct.

Note: Note that matrix multiplication is not commutative i.e. $P\times Q$ will not be the same as $Q\times P$, so make sure that you look at the orders properly. Like in this case, if the orders would have been the other way round, then matrix multiplication would not have been possible.