Q1.3 30E In problems 29 through 32, let聽... [FREE SOLUTION]

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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: Q1.3 30E (page 35)

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

Short Answer

Expert verified

The matrix A of rank 1 such that $A \overset{鈫�}{x} = \overset{鈫�}{y}$ is, role="math" localid="1664193119490" $[\begin{matrix} \frac{2}{5} & 0 & 0 \\ 0 & 0 & 0 \\ \frac{1}{5} & 0 & 0 \end{matrix}]$ .

Step by step solution

Consider the matrices

The rank of a matrix A is the number of leading 1鈥檚 in rref(A), denoted as rank(A).

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

The matrix A of rank 1 is,role="math" localid="1664193081786" $[\begin{matrix} x & 0 & 0 \\ y & 0 & 0 \\ z & 0 & 0 \end{matrix}]$

Perform the multiplication operation.

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

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

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

The values are:

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

Compute the matrix.

Let the matrix A of rank 1 is, $[\begin{matrix} x & 0 & 0 \\ y & 0 & 0 \\ z & 0 & 0 \end{matrix}]$ .

Substitute the values of x, y and z.

role="math" localid="1664193133621" $[\begin{matrix} x & 0 & 0 \\ y & 0 & 0 \\ z & 0 & 0 \end{matrix}] = [\begin{matrix} \frac{2}{5} & 0 & 0 \\ 0 & 0 & 0 \\ \frac{1}{5} & 0 & 0 \end{matrix}]$

Hence, $[\begin{matrix} \frac{2}{5} & 0 & 0 \\ 0 & 0 & 0 \\ \frac{1}{5} & 0 & 0 \end{matrix}]$ is the matrix of A with rank 1 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 $\overset{鈫�}{y} = [\begin{matrix} 2 \\ 0 \\ 1 \end{matrix}]$ .
30. Find a matrix A of rank 1 such that $A \overset{鈫�}{x} = \overset{鈫�}{y}$ .

Short Answer

Step by step solution

Consider the matrices

Perform the multiplication operation.

Compute the matrix.

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91影视

In problems 29 through 32, letx鈫�=[53-9] and y鈫�=[201].30. Find a matrix A of rank 1 such that Ax鈫�=y鈫�.

Short Answer

Step by step solution

Consider the matrices

Perform the multiplication operation.

Compute the matrix.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

Statistics

Geometry

Logic and Functions

Study anywhere. Anytime. Across all devices.

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