Question

Two rectangular matrices of order $n\times m$ and $m\times k$ are multiplied in the same order. The resulting matrix formed is a
This question has multiple correct options
A. Rectangular Matrix of order $n\times k$
B. Square matrix of order m
C. Square matrix of order n if n = k
D. Rectangular Matrix of order $k\times n$

Answer 1

Hint: If there are m rows and n columns in the matrix, then the matrix is called a rectangular matrix of order $m\times n$ and a matrix having equal number of rows and columns is called square matrix of order n.

Complete step by step answer:
Let us consider the example,
Let A be the matrix of order $3\times 2$ and B be another matrix of order $2\times 1$, where $A={{\left[ \begin{matrix} 1 \\\ 3 \\\ 1 \\\ \end{matrix}\text{ }\begin{matrix} 2 \\\ 0 \\\ 1 \\\ \end{matrix} \right]}_{3\times 2}}$ and $B={{\left[ \begin{matrix} 2 \\\ 1 \\\ \end{matrix} \right]}_{2\times 1}}$
Now, $AB={{\left[ \begin{matrix} 1 \\\ 3 \\\ 1 \\\ \end{matrix}\text{ }\begin{matrix} 2 \\\ 0 \\\ 1 \\\ \end{matrix} \right]}_{3\times 2}}{{\left[ \begin{matrix} 2 \\\ 1 \\\ \end{matrix} \right]}_{2\times 1}}={{\left[ \begin{matrix} 2+2 \\\ 3+0 \\\ 2+1 \\\ \end{matrix} \right]}_{3\times 1}}={{\left[ \begin{matrix} 4 \\\ 3 \\\ 3 \\\ \end{matrix} \right]}_{3\times 1}}$
Two rectangular matrices of order $3\times 2$ and $2\times 1$ are multiplied, then the resulting matrix is a rectangular matrix of the order $3\times 1$.

Hence multiplying two rectangular matrices of order $n\times m$ and $m\times k$ is a rectangular matrix of order $n\times k$.

Therefore, the correct option is option (a).

Also, consider another example,
Let C be the matrix of order $2\times 3$ and D be another matrix of order $3\times 2$, where $C={{\left[ \begin{matrix} 2 \\\ 1 \\\ \end{matrix}\text{ }\begin{matrix} 0 & 1 \\\ 1 & 2 \\\ \end{matrix} \right]}_{2\times 3}}$ and $D={{\left[ \begin{matrix} 2 \\\ 1 \\\ 1 \\\ \end{matrix}\text{ }\begin{matrix} 0 \\\ 3 \\\ 1 \\\ \end{matrix} \right]}_{3\times 2}}$

Now, $CD={{\left[ \begin{matrix} 2 \\\ 1 \\\ \end{matrix}\text{ }\begin{matrix} 0 & 1 \\\ 1 & 2 \\\ \end{matrix} \right]}_{2\times 3}}{{\left[ \begin{matrix} 2 \\\ 1 \\\ 1 \\\ \end{matrix}\text{ }\begin{matrix} 0 \\\ 3 \\\ 1 \\\ \end{matrix} \right]}_{3\times 2}}={{\left[ \begin{matrix} 4+0+1 \\\ 2+1+2 \\\ \end{matrix}\text{ }\begin{matrix} 0+0+1 \\\ 0+3+2 \\\ \end{matrix} \right]}_{2\times 2}}={{\left[ \begin{matrix} 5 \\\ 5 \\\ \end{matrix}\text{ }\begin{matrix} 1 \\\ 5 \\\ \end{matrix} \right]}_{2\times 2}}$
Two rectangular matrices of order $2\times 3$ and $3\times 2$ are multiplied, then the resulting matrix is a square matrix of the order $2\times 2$.

Hence multiplying two rectangular matrices of order $n\times m$ and $m\times k$ is a square matrix of order n if n = k.

Therefore, the correct option is option (c).

Hence correct options for the given question are option (A) and option (C).

Note: The possibility for the mistake is that you might get confused with the concept that two matrices A and B are said to be conformable for the product AB if the number of columns in A is equal to the number of rows in B.