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Engineering Mathematics

Row Echelon Form

The rank of t...

Question

The rank of the matrix $⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 & 0 & 0 0 & 0 & 1 & - 1 & 0 0 & 1 & - 1 & 0 & 0 - 1 & 0 & 0 & 0 & 1 0 & 0 & 0 & 1 & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$ is ____.
4

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Solution

The correct option is A 4
$\begin{matrix} r_{1} r_{2} r_{3} r_{4} r_{5} \end{matrix} ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & - 1 & 0 & 0 & 0 0 & 0 & 1 & - 1 & 0 0 & 1 & - 1 & 0 & 0 - 1 & 0 & 0 & 0 & 1 0 & 0 & 0 & 1 & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

$r_{1} \to r_{1} + r_{2} + r_{3} + r_{4} + r_{5}$

We get,

$A = {⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & 0 & 0 & 0 & 0 0 & 0 & 1 & - 1 & 0 0 & 1 & - 1 & 0 & 0 - 1 & 0 & 0 & 0 & 1 0 & 0 & 0 & 1 & - 1 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦}_{5 \times 5}$

So, $| A | = 0$
Hence rank(A)<5

But now another row is not becoming zero.
So at least one 4 $\times$ 4 matrix in A is not zero
Hence Rank = 4

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