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Q14E

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Found in: Page 39

### Linear Algebra and its Applications

Book edition 5th
Author(s) David C. Lay, Steven R. Lay and Judi J. McDonald
Pages 483 pages
ISBN 978-03219822384

# Question: Determine whether the statements that follow are true or false, and justify your answer.14: rank .$|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 6\end{array}|{\mathbf{=}}{\mathbf{3}}$

True, rank of the given matrix is 3 because after finding the row echelon form there are three non-zero rows.

See the step by step solution

## Step 1: Rank of a matrix

For finding the rank of the matrix first of all we will change the given matrix in the echelon form.

We have given the matrix.

$\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 6\end{array}\right|$

## Step 2: Justification of answer

Now, find the reduced row echelon form of the given system.

$\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 3& 6\end{array}\right|\phantom{\rule{0ex}{0ex}}{R}_{2}\to {R}_{2}-{R}_{1}\phantom{\rule{0ex}{0ex}}{R}_{3}\to {R}_{3}-{R}_{1}\phantom{\rule{0ex}{0ex}}~\left|\begin{array}{ccc}1& 1& 1\\ 0& 1& 2\\ 0& 2& 5\end{array}\right|\phantom{\rule{0ex}{0ex}}{R}_{3}\to {R}_{3}-2{R}_{2}\phantom{\rule{0ex}{0ex}}~\left|\begin{array}{ccc}1& 1& 1\\ 0& 1& 2\\ 0& 0& 1\end{array}\right|$

Now, the given matrix is in echelon form. Thus, the number of non-zero rows is equal to the rank of a matrix.

Thus, the rank of a matrix is 3.

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