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)

1

Answer 2

Let A = $\begin{bmatrix}a_{11}\\\ a_{21}\\\ \\\ a_{m1}\end{bmatrix}$ and B = $[b_{11} \, b_{12}\,b_{13} ..... b_{1n} ]$ be two non-zero column and row matrices respectively $\begin{bmatrix}a_{11}b_{11}&a_{11}b_{12}&a_{11}b_{13}&...&a_{11}b_{1n}\\\ a_{21}b_{11}&a_{21}b_{12}&a_{21}b_{13}&...&a_{21}b_{1n}\\\ .....&.....&.....&...&.....\\\ a_{m1}b_{11}&a_{m1}b_{12}&a_{m1}b_{13}&...&a_{m1}b_{1n}\end{bmatrix}$ 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