Q52E Consider two vectors聽v鈬赌 and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 6: Q52E (page 276)

Consider two vectors $\overset{鈬赌}{v}$ and $\overset{鈬赌}{w}$ in ${鈩�}^{3}$ . Form the matrix $A = [\overset{鈬赌}{v} x \overset{鈬赌}{w} \overset{鈬赌}{v} \overset{鈬赌}{w}]$ . Express detA in terms of $|| \overset{鈬赌}{v} x \overset{鈬赌}{w} ||$ . For which choices of $\overset{鈬赌}{v}$ and $\overset{鈬赌}{w}$ is Ainvertible?

Short Answer

Expert verified

$d e t [\overset{鈬赌}{v} x \overset{鈬赌}{w} \overset{鈬赌}{v} \overset{鈬赌}{w}] = {||\overset{鈬赌}{v} x \overset{鈬赌}{w}||}^{2}$

Therefore, the matrix is invertible if and only $\overset{鈬赌}{v}$ if $\overset{鈬赌}{w}$ and are non-parallel.

Step by step solution

Matrix Definition.

Matrix is a set of numbers arranged in rows and columns so as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are m rows and n columns, the matrix is said to be an 鈥渕 by n鈥� matrix, written 鈥渕xn鈥�

To find which choices of matrix is invertible.

To solve,

$d e t [\overset{鈬赌}{v} x \overset{鈬赌}{w} \overset{鈬赌}{v} \overset{鈬赌}{w}] = [\overset{鈬赌}{v} x \overset{鈬赌}{w}, \overset{鈬赌}{v} x \overset{鈬赌}{w}] = {||\overset{鈬赌}{v} x \overset{鈬赌}{w}||}^{2}$

For any given $\overset{鈬赌}{v}, \overset{鈬赌}{w} 鈭� {鈩�}^{3}$ , the vector $\overset{鈬赌}{v} x \overset{鈬赌}{w}$ is perpendicular to both vectors, meaning it's never parallel to either one of them, so the given matrix is invertible if and only if $\overset{鈬赌}{v} and \overset{鈬赌}{w}$ are non-parallel.

Therefore, the matrix is invertible if and only if $\overset{鈬赌}{v} and \overset{鈬赌}{w}$ are non-parallel.

$d e t [\overset{鈬赌}{v} x \overset{鈬赌}{w} \overset{鈬赌}{v} \overset{鈬赌}{w}] = {||\overset{鈬赌}{v} x \overset{鈬赌}{w}||}^{2}$

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Short Answer

Step by step solution

Matrix Definition.

To find which choices of matrix is invertible.

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Consider two vectorsv鈬赌 and w鈬赌in鈩�3. Form the matrix A=[v鈬赌xw鈬赌v鈬赌w鈬赌] . Express detA in terms of||v鈬赌xw鈬赌||. For which choices of v鈬赌 and w鈬赌 is Ainvertible?

Short Answer

Step by step solution

Matrix Definition.

To find which choices of matrix is invertible.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Discrete Mathematics

Logic and Functions

Applied Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.