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Elementary Transformations

Find the rank...

Question

Find the rank of the matrix $A = ⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 9 4 & 8 & 12 7 & 11 & 15 \end{matrix} ⎤ ⎥ ⎦$

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Solution

Minor of highest order 3 is $⎡ ⎢ ⎣ \begin{matrix} 1 & 5 & 9 4 & 8 & 12 7 & 11 & 15 \end{matrix} ⎤ ⎥ ⎦ = 0$ by $R_{1} + R_{3} - 2 R_{2}$
Minor of order 2 is $[\begin{matrix} 1 & 5 4 & 8 \end{matrix}] = 8 - 20 = - 12 \neq 0$
Hence $ρ (A) = 2$

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