Question

If v is a non-zero vector of dimension 3 × 1, then the matrix A = vv^T has a rank = _______

Answer 1

v is a non-zero vector of dimension 3 × 1

Let the matrix \(v = \left[ {\begin{array}{*{20}{c}} a\\ b\\ c \end{array}} \right]\)

\({v^T} = \left[ {\begin{array}{*{20}{c}} a&b&c \end{array}} \right]\)

\(A = v{v^T} = \left[ {\begin{array}{*{20}{c}} a\\ b\\ c \end{array}} \right]\left[ {\begin{array}{*{20}{c}} a&b&c \end{array}} \right]\)

\(= \left[ {\begin{array}{*{20}{c}} {{a^2}}&{ab}&{ac}\\ {ab}&{{b^2}}&{bc}\\ {ac}&{bc}&{{c^2}} \end{array}} \right]\)

\(= abc\left| {\begin{array}{*{20}{c}} a&b&c\\ a&b&c\\ a&b&c \end{array}} \right| = {a^2}{b^2}{c^2}\left| {\begin{array}{*{20}{c}} 1&1&1\\ 1&1&1\\ 1&1&1 \end{array}} \right|\)

As third order and second order determinants are zero, the rank of the matrix is 1

Alternate approach:

From the properties of matrix

\(\rho \left( {AB} \right) \le \min \left\{ {\rho \left( A \right),\;\rho \left( B \right)} \right\}\)

Rank of matrix v = ρ(v) = 1

Rank pf matrix v^T = ρ(v^T) = 1

ρ(A = vv^T) ≤ min (ρ(v) , ρ(v^T) )

≤ min (1, 1)

Rank of A = ρ(A) = 1