Question

If A is a non-zero column matrix of order m x 1 and B is non-zero row matrix of order $1\times n$, then rank of AB is equal to

• A. m
• B. n
• C. 1
• D. none of these.

Answer 1

Answer: (C)

Let
A = $\left[\begin{array}{c}{a}_{11}\\ a_{21}\\ \\ a_{m1}\end{array}\right]$ and B = $[b_{11}{b}_{12}{b}_{13}.....b_{1n}]$
be two non-zero column and row matrices respectively
$\left[\begin{array}{ccccc}{a}_{11}{b}_{11}& a_{11}{b}_{12}& a_{11}{b}_{13}& ...& a_{11}{b}_{1n}\\ a_{21}{b}_{11}& a_{21}{b}_{12}& a_{21}{b}_{13}& ...& a_{21}{b}_{1n}\\ .....& .....& .....& ...& .....\\ a_{m1}{b}_{11}& a_{m1}{b}_{12}& a_{m1}{b}_{13}& ...& a_{m1}{b}_{1n}\end{array}\right]$
Since A, B are non-zero matrices. $\therefore$ matrix AB will be a non-zero matrix. The matrix AB will have at least one non-zero element obtained by multiplying corresponding non-zero elements of A and B. All the two rowed minors of AB clearly vanish. Since AB is non-zero matrix,
$\therefore$ rank of AB = 1