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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q17E (page 1)

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

Short Answer

Expert verified

False, Rank of $|\begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix}| = 1$

Step by step solution

01

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.

$|\begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix}|$

02

Justification of answer

Now, reduce the given matrix into row echelon form.

$|\begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix}| R_{2} 鈫� R_{2} - R_{1} R_{3} 鈫� R_{3} - R_{1} ~ |\begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix}|$

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.

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$\begin{array}{l} \overset{鈫�}{b} = [\begin{matrix} 13 .2 \\ 17 .6 \\ 1 .8 \end{matrix}], \overset{鈫�}{蠀_{1}} = [\begin{matrix} 0 .293 \\ 0 .014 \\ 0 .044 \end{matrix}], \\ \overset{鈫�}{蠀_{2}} = [\begin{matrix} 0 \\ 0 .207 \\ 0 .01 \end{matrix}], \overset{鈫�}{蠀_{3}} = [\begin{matrix} 0 \\ 0 .017 \\ 0 .216 \end{matrix}] \end{array}$

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