Question

If A is a m × n matrix such that AB and BA are both defined, then order of B is
$\begin{align}
  & a)m\times m \\
 & b)n\times m \\
 & c)n\times n \\
 & d)m\times m \\
\end{align}$

Answer 1

Hint: Now let us first understand matrices.
Now matrices are basically tabular structure with rows and columns.
Now let us say our matrix has m rows and n columns. Then we say that the matrix is of order m × n.
For example consider matrix A such that \[A=\left[ \begin{matrix}
   1 \\
   1 \\
\end{matrix}\begin{matrix}
   3 \\
   2 \\
\end{matrix}\begin{matrix}
   2 \\
   5 \\
\end{matrix} \right]\] Now we can see that the matrix has 2 rows and 3 columns hence order of matrix A is 2 × 3.
Now let us understand the condition for multiplication of matrices
Hence Let us say we want to multiply Matrix A and B. Then for the multiplication to be defined number of columns of A should be equal to the number of rows of B.
Hence if the order of A is m × n then the order of B should be n × p.

Complete step by step answer:
Now let us consider the Matrix A with order m × n and B with order p × q.
Let us say AB is defined
Then the number of columns of A is equal to the number of rows of B
Hence we get n = p …………………. (1)
Now let us say BA is also defined.
Then we have the number of columns of B is equal to the number of rows of A.
Hence we get q = m …………………… (2)
Now from equation (1) and equation (2) we get
p × q = n × m.
Hence the order of matrix B must be n × m.
Option c is the correct option.
Note: note that matrix multiplication is defined in a complex manner. It is not just multiplying each respective element of both matrices like addition. Now when we multiply an m × n matrix with n × p matrix we get m × p matrix.