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Q17E

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Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

Determine whether the statements that follow are true or false, and justify your answer.17: Rank $|\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}|{\mathbf{=}}{\mathbf{2}}$

False, Rank of $\left|\begin{array}{ccc}2& 2& 2\\ 2& 2& 2\\ 2& 2& 2\end{array}\right|=1$

See the step by step solution

Step 1:Rank of matrix

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

We have given the matrix.

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

Step 2: Justification of answer

Now, reduce the given matrix into row echelon form.

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

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

Thus, the rank of the given matrix is 1.