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# If A=$\begin{bmatrix} 1\\ 2 \\ 3 \end{bmatrix}$ than the rank of $AA^{T}$is

$\begin{array}{1 1}(1) 3&(2) 0\\(3)1&(4) 2\end{array}$

A=$\begin{bmatrix} 1\\ 2 \\ 3 \end{bmatrix}$
$A^T=\begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$
$AA^T=\begin{bmatrix} 1\\ 2 \\ 3 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}$
$\qquad=\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{bmatrix}$
$\qquad=\begin{bmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$
$R_2 \to R_2 - 2R_1$
$R_3 \to R_3 - 3R_1$
The No. of non zero rows in $=1$
$p(A)=1$
Hence 1 is the correct answer.
edited May 2, 2014 by meena.p