Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 6: Q10E (page 306)

Consider a $n 脳 n$ matrix $A = [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{1} . . . . {\overset{鈫�}{v}}_{n}]$ . What is the relationship between the product $|| {\overset{鈫�}{v}}_{1} || || {\overset{鈫�}{v}}_{2} || . . . || {\overset{鈫�}{v}}_{n} ||$ and $| detA |$ When is $| detA | = || {\overset{鈫�}{v}}_{1} || || {\overset{鈫�}{v}}_{2} || . . . || {\overset{鈫�}{v}}_{n} ||$ .

Short Answer

Expert verified

The equality holds if and only if $||{\overset{鈫�}{v}}_{i}^{鈯�}|| = ||{\overset{鈫�}{v}}_{i}^{}||, 鈭赌 i 鈭� \{1, . . ., n\}$ , which is if and only if all the vectors are mutually perpendicular.

Step by step solution

Matrix Definition.

Matrix is aset of numbersarranged in rows and columns so asto form a rectangulararray.

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

If there are rows and columns, the matrix is said to be a 鈥� m by n 鈥� matrix, written 鈥� $m 脳 n$ .鈥�

To find the relationship between the product ||v→1||||v→2||...||v→n|| and |detA|

We know that,

$|d e t A| = ||\overset{鈫�}{v_{1}}|| ||{\overset{鈫�}{v}}_{2}^{鈯�}|| . . . . ||{\overset{鈫�}{v}}_{n}^{鈯�}||$

Since $||{\overset{鈫�}{v}}_{i}^{鈯�}|| = ||{\overset{鈫�}{v}}_{i}^{}||, 鈭赌 i 鈭� \{1, . . ., n\}$ , then $|d e t A| 鈮� ||{\overset{鈫�}{v}}_{1}|| ||{\overset{鈫�}{v}}_{2}|| . . . . . ||{\overset{鈫�}{v}}_{n}||$ .

The equality holds if and only if $||{\overset{鈫�}{v}}_{i}^{鈯�}|| = ||{\overset{鈫�}{v}}_{i}^{}||, 鈭赌 i 鈭� \{1, . . ., n\}$ , which is if and only if all the vectors ${\overset{鈫�}{v}}_{i}$ are mutually perpendicular.

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91影视

Short Answer

Step by step solution

Matrix Definition.

To find the relationship between the product ||v→1||||v→2||...||v→n|| and |detA|

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91影视

Consider a n脳n matrixA=[v鈫�1v鈫�1....v鈫�n] . What is the relationship between the product ||v鈫�1||||v鈫�2||...||v鈫�n|| and |detA|When is|detA|=||v鈫�1||||v鈫�2||...||v鈫�n|| .

Short Answer

Step by step solution

Matrix Definition.

To find the relationship between the product ||v→1||||v→2||...||v→n|| and |detA|

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Discrete Mathematics

Mechanics Maths

Pure Maths

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.