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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: Q4E (page 34)

Find the rank of the matrices in 2 through 4.
4. $[\begin{matrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{matrix}]$

Short Answer

Expert verified

The rank of the matrix $[\begin{matrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{matrix}]$ is, $2$ .

Step by step solution

01

Find row reduce echelon form

The number of leading 1鈥檚 in $rref (A)$ represents the rank of a matrix A denoted by $rank (A)$ .

The given matrix is, $[\begin{matrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{matrix}]$ .

The reduced row echelon form of the given matrix is:

$r r e f [\begin{matrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{matrix}] = [\begin{matrix} 1 & 0 & 鈭� 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{matrix}]$

02

Determine the rank of a matrix.

The number of leading 1鈥檚 in the matrix $[\begin{matrix} 1 & 0 & 鈭� 1 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{matrix}]$ are $2$ .

Hence, the rank of the given matrix is 2.

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Most popular questions from this chapter

Compute the products Ax in Exercises 13 through 15 using

paper and pencil. In each case, compute the product

two ways: in terms of the columns of A and in terms of the rows of A.

15.

$[1 2 3 4] [\begin{matrix} 5 \\ 6 \\ 7 \\ 8 \end{matrix}]$

in exercises 1 through 10, find all solutions of the linear systems using elimination.Then check your solutions

$|x + 2 y = 1 2 x + 3 y = 1|$

Let be an orthogonal 2X2 matrix. Use the image of the unit circle to find the singular values of A.

Consider a solution $\overset{鈫�}{x_{1}}$ of the linear system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ . Justify the facts stated in parts (a) and (b):

a. If ${\overset{鈫�}{x}}_{h}$ is a solution of the system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ , then $\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{h}}$ is a solution of the system $A = \overset{鈫�}{x} = \overset{鈫�}{b}$ .

b. If $\overset{鈫�}{x_{2}}$ is another solution of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ , then $\overset{鈫�}{x_{1}} + \overset{鈫�}{x_{h}}$ is a solution of the system $A \overset{鈫�}{x} + \overset{鈫�}{0}$ .

c. Now suppose A is a $2 脳 2$ matrix. A solution vector $\overset{鈫�}{x_{1}}$ of the system $A \overset{鈫�}{x} + \overset{鈫�}{b}$ is shown in the accompanying figure. We are told that the solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{0}$ form the line shown in the sketch. Draw the line consisting of all solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ .

If you are puzzled by the generality of this problem, think about an example first:

$A = (\begin{array}{ll} 1 鈥呪赌呪赌呪赌� 2 \\ 3 鈥呪赌呪赌呪赌� 6 \end{array}), \overset{鈫�}{b} = [\begin{array}{l} 3 \\ 9 \end{array}] and \overset{鈫�}{x_{1}} = [\begin{array}{l} 1 \\ 1 \end{array}]$

Kyle is getting some flowers for Olivia, his Valentine. Being of a precise analytical mind, he plans to spend exactly \(24 on a bunch of exactly two dozen flowers. At the flower market they have lilies (\)3 each), roses (\(2 each), and daisies (\)0.50 each). Kyle knows that Olivia loves lilies; what is he to do?

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