Question

If A is a non-zero column matrix of order \[m\times 1\] and B is a non-zero row matrix of order \[\text{1xn}\] then rank of AB equals?
A. m
B. n
C. 1
D. None of these.

Answer 1

Hint:First multiply the given matrices \[A\left( m\times 1 \right)\text{ and B}\left( 1\times n \right)\] to find the product AB matrix, then, find the rank of AB matrix.

Complete step by step answer:
First of all, we must know what a matrix is, matrix is a rectangular or square arrangement of number into rows and columns. It can be 1 row 2 column, 2 rows 3 columns, etc.
The rank of any matrix is given as the maximum number of linearly independent column or row vectors in the matrix.
Suppose, we have to matrix A having 2 rows 3 columns, then, we will write it as:
\[\begin{align}
  & A=\text{ }\underbrace{\left. \left[ \begin{matrix}
   a \\
   b \\
\end{matrix}\text{ }\begin{matrix}
   b \\
   e \\
\end{matrix}\text{ }\begin{matrix}
   c \\
   f \\
\end{matrix} \right] \right\}}_{Column}\text{ Rows} \\
 & \text{A=2}\times \text{3 Matrix} \\
 & \text{ } \\
\end{align}\]
In the question, we are given 2 matrices A and B.
\[A=\left( m\times 1 \right)\to \text{ Column matrix}\]
i.e., A = it has only 1 column.
\[A=\left[ \begin{matrix}
   a \\
   b \\
   c \\
   \begin{align}
  & ' \\
 & ' \\
\end{align} \\
\end{matrix} \right]\]
\[B=\left( 1\times n \right)\to \text{ Row matrix}\]
i.e., B = it has only 1 row.
\[B=\left[ \begin{matrix}
   a & b & c\text{ - -} \\
\end{matrix} \right]\]
We are given non-zero matrices A and B. This means there is at least one non-zero element in the matrix. For simplification purposes, we will consider matrices with one element as 1 and rest of the elements as as 0.
\[\Rightarrow A={{\left[ \begin{matrix}
   1 \\
   0 \\
   0 \\
   \begin{align}
  & ' \\
 & ' \\
\end{align} \\
\end{matrix} \right]}_{m\times 1}}\]
\[\Rightarrow B={{\left[ \begin{matrix}
   1 & 0 & 0 & 0 \\
\end{matrix}\text{ - -} \right]}_{1\times n}}\]
Now, find the product of A and B \[\Rightarrow A\times B=AB\]
Order of AB = rows of A and columns of B.
\[\therefore AB={{\left[ \begin{matrix}
   1 & 0 & 0 \\
   0 & 0 & 0 \\
   ' & ' & ' \\
\end{matrix} \right]}_{m\times n}}\]
Rank of AB will be number of independent column or row vectors in the matrix. Only the first row and first column have a non-zero element. Hence, the rank of AB is 1.
So, option C is correct.

Note:
It is not necessary to assume the matrices A and B to be containing elements only 1 and rest zero. Students can assume any row and column matrices, but make sure that should not get lengthy and complex.