Question

The rank of a 3 $\times$ 3 matrix C (= AB), found by multiplying a non-zero column matrix A of size 3 $\times$ 1 and a non-zero row matrix B of size 1 $\times$ 3, is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

The correct option is B 1

Let A = $\left[\begin{array}{c}{a}_1\\ b_1\\ c_1\end{array}\right]$ & B = $\left[\begin{array}{ccc}{a}_2& b_2& c_2\end{array}\right]$

rank of A = r(A) = 1
rank of B = r(B) = 1

Rank of (AB) $\le$ minimum (Rank (A), Rank (B))

r(AB) $\le$ 1

So, rank (AB) is either 0 or 1

$\because ButAB\ne 0$

$\because Rank\left(AB\right)=1$