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Q.
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
Matrices
Solution:
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