then the rank of AAT is

Engineering

Mathematics

Multiplication of Matrices

Symmetric and Skew Symmetric Matrices

Question

$A = [\begin{array}{c} 1 \\ 2 \\ 3 \end{array}],$ then the rank of AA^T is

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$A = {[\begin{array}{l} 1 \\ 2 \\ 3 \end{array}]}_{3 \times 1}$ [1 2 3]_1×3

$= {[\begin{array}{lll} 1 2 3 \\ 2 4 6 \\ 3 6 9 \end{array}]}_{3 \times 3}$

we reduce M in the echelon form.

$M = [\begin{array}{lll} 1 2 3 \\ 2 4 6 \\ 3 6 9 \end{array}]$

$\begin{aligned} R_{2} \to R_{2} - 2 R_{1} \\ R_{3} \to R_{3} - 3 R_{1} \end{aligned} : [\begin{array}{lll} 1 2 3 \\ 0 0 0 \\ 0 0 0 \end{array}]$

∴ rank of M = 1.

i.e. rank of AAT = 1.

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