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Engineering Mathematics

Elementary row Operations

The rank of t...

Question

The rank of the matrix $⎡ ⎢ ⎣ \begin{matrix} 6 & 0 & 4 & 4 - 2 & 14 & 8 & 18 14 & - 14 & 0 & - 10 \end{matrix} ⎤ ⎥ ⎦$ is ______ .
2

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Solution

The correct option is A 2
Let $A = ⎡ ⎢ ⎣ \begin{matrix} 6 & 0 & 4 & 4 - 2 & 14 & 8 & 18 14 & - 14 & 0 & - 10 \end{matrix} ⎤ ⎥ ⎦$

$R_{1} \leftrightarrow R_{2}$

$A \sim ⎡ ⎢ ⎣ \begin{matrix} - 2 & 14 & 8 & 18 6 & 0 & 4 & 4 14 & - 14 & 0 & - 10 \end{matrix} ⎤ ⎥ ⎦$

$R_{2} \to R_{2} + 3 R_{1}; R_{3} \to R_{3} + 7 R_{1}$

$A \sim ⎡ ⎢ ⎣ \begin{matrix} - 2 & 14 & 8 & 18 0 & 42 & 28 & 58 0 & 84 & 56 & 116 \end{matrix} ⎤ ⎥ ⎦$

$R_{3} \to R_{3} - 2 R_{2}$

$A \sim ⎡ ⎢ ⎣ \begin{matrix} - 2 & 14 & 8 & 18 0 & 42 & 28 & 58 0 & 0 & 0 & 0 \end{matrix} ⎤ ⎥ ⎦$

= Echelon form
Number of non - zero rows = 2

So $ρ (A) = 2$

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