Q1.3 29E In problems 29 through 32, let聽... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 1: Q1.3 29E (page 35)

In problems 29 through 32, let $\overset{鈫�}{x} = [\begin{matrix} 5 \\ 3 \\ - 9 \end{matrix}]$ and $\overset{鈫�}{y} = [\begin{matrix} 2 \\ 0 \\ 1 \end{matrix}]$ .
29. Find a diagonal matrix A such that $A \overset{鈫�}{x} = \overset{鈫�}{y}$ .

Short Answer

Expert verified

A diagonal matrix A such that $A \overset{鈫�}{x} = \overset{鈫�}{y}$ is, localid="1664191727343" $[\begin{matrix} \frac{2}{5} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - \frac{1}{9} \end{matrix}]$

Step by step solution

Consider the matrices

A diagonal matrix is a matrix with non-zero diagonal elements and all other zero elements. $(\begin{matrix} a_{1} & 鈥� & 0 \\ 鈰� & 鈰� & 鈰� \\ 0 & 鈥� & a_{n} \end{matrix})$ .

The matrices are, $\overset{鈫�}{x} = [\begin{matrix} 5 \\ 3 \\ - 9 \end{matrix}]$ and $\overset{鈫�}{y} = [\begin{matrix} 2 \\ 0 \\ 1 \end{matrix}]$ .

The diagonal matrix A is, localid="1664191671433" role="math" $[\begin{matrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{matrix}]$

Perform the multiplication operation.

The system $A \overset{鈫�}{x} = \overset{鈫�}{y}$ is,

$[\begin{matrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{matrix}] [\begin{matrix} 5 \\ 3 \\ - 9 \end{matrix}] = [\begin{matrix} 2 \\ 0 \\ 1 \end{matrix}]$

$[\begin{matrix} 5 x \\ 3 y \\ - 9 z \end{matrix}] = [\begin{matrix} 2 \\ 0 \\ 1 \end{matrix}]$

The values are:

$x = \frac{2}{5}, y = 0, z = - \frac{1}{9}$

Compute the diagonal matrix.

The diagonal matrix A is, $[\begin{matrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{matrix}]$

Substitute the values of x, y and z.

role="math" localid="1664191740773" $[\begin{matrix} x & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & z \end{matrix}] = [\begin{matrix} \frac{2}{5} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - \frac{1}{9} \end{matrix}]$

Hence, $[\begin{matrix} \frac{2}{5} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & - \frac{1}{9} \end{matrix}]$ is the diagonal matrix of A such that $A \overset{鈫�}{x} = \overset{鈫�}{y}$ .

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91影视

In problems 29 through 32, let $\overset{鈫�}{x} = [\begin{matrix} 5 \\ 3 \\ - 9 \end{matrix}]$ and $\overset{鈫�}{y} = [\begin{matrix} 2 \\ 0 \\ 1 \end{matrix}]$ .
29. Find a diagonal matrix A such that $A \overset{鈫�}{x} = \overset{鈫�}{y}$ .

Short Answer

Step by step solution

Consider the matrices

Perform the multiplication operation.

Compute the diagonal matrix.

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91影视

In problems 29 through 32, letx鈫�=[53-9] and y鈫�=[201].29. Find a diagonal matrix A such that Ax鈫�=y鈫�.

Short Answer

Step by step solution

Consider the matrices

Perform the multiplication operation.

Compute the diagonal matrix.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Calculus

Mechanics Maths

Decision Maths

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

In problems 29 through 32, let $\overset{鈫�}{x} = [\begin{matrix} 5 \\ 3 \\ - 9 \end{matrix}]$ and $\overset{鈫�}{y} = [\begin{matrix} 2 \\ 0 \\ 1 \end{matrix}]$ .
29. Find a diagonal matrix A such that $A \overset{鈫�}{x} = \overset{鈫�}{y}$ .