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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: Q9E (page 35)

Write the system in matrix form.
$| \begin{matrix} x + 2 y + 3 z = 1 \\ 4 x + 5 y + 6 z = 4 \\ 7 x + 8 y + 9 z = 9 \end{matrix} |$

Short Answer

Expert verified

The matrix form

$A \overset{鈫�}{x} = \overset{鈫�}{b} i s [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 4 \\ 9 \end{matrix}]$

Step by step solution

01

Step 1:

The matrix of the coefficients a are

$A = [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}]$

The matrix with the variables is-

$\overset{鈫�}{x} = [\begin{matrix} x \\ y \\ z \end{matrix}]$

The matrix of constants is

$[\begin{matrix} 1 \\ 4 \\ 9 \end{matrix}]$

Hence, the matrix form

$A \overset{鈫�}{x} = \overset{鈫�}{b} i s [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 4 \\ 9 \end{matrix}]$

02

Final answer:

The matrix form

$A \overset{鈫�}{x} = \overset{鈫�}{b} i s [\begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 1 \\ 4 \\ 9 \end{matrix}]$

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Most popular questions from this chapter

Find the rank of the matrices in 2 through 4.

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

Consider the equations

$| \begin{matrix} y + 2 k z = 0 \\ x + 2 y + 6 z = 2 \\ k x + 2 z = 1 \end{matrix} |$

where $k$ is an arbitrary constant.

a. For which values of the constant $k$ does this system have a unique solution?

b. When is there no solution?

c. When are there infinitely many solutions?

Consider the economy of Israel in 1958. The three industries considered here are

$\begin{array}{l} I_{1} : 鈥塧驳谤颈肠耻濒迟耻谤别, \\ I_{2} : 鈥尘补苍耻蹿补肠迟耻谤颈苍驳, \\ I_{3} : 鈥塭苍别谤驳测 \end{array}$

Outputs and demands are measured in millions of Israeli pounds, the currency of Israel at that time. We are told that

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

a. Why do the first components of $\overset{鈫�}{蠀_{2}}$ and $\overset{鈫�}{蠀_{3}}$ equal $0$ ?

b. Find the outputs $x_{1}$ , $x_{2}$ , $x_{3}$ required to satisfy demand.

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

3. $|2 x + 4 y = 3 3 x + 6 y = 2|$

Compute the products Axin 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.

14.

$[\begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 4 \end{matrix}] [\begin{matrix} - 1 \\ 2 \\ 1 \end{matrix}]$

See all solutions

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