×
Get Full Access to Linear Algebra And Its Applications - 5 Edition - Chapter 2.4 - Problem 24e
Get Full Access to Linear Algebra And Its Applications - 5 Edition - Chapter 2.4 - Problem 24e

×

# Use partitioned matrices to prove by induction that for n

ISBN: 9780321982384 49

## Solution for problem 24E Chapter 2.4

Linear Algebra and Its Applications | 5th Edition

• Textbook Solutions
• 2901 Step-by-step solutions solved by professors and subject experts
• Get 24/7 help from StudySoup virtual teaching assistants

Linear Algebra and Its Applications | 5th Edition

4 5 1 398 Reviews
24
0
Problem 24E

Problem 24E

Use partitioned matrices to prove by induction that for n = 2, 3,…,the n × n matrix A shown below is invertible and B is its inverse.

For the induction step, assume A and B are (k + 1) × (k + 1) matrices, and partition A and B in a form similar to that displayed in Exercise 23.

Exercise 23:

Use partitioned matrices to prove by induction that the product of two lower triangular matrices is also lower triangular. [Hint: A(k + 1) × (k + 1) matrix A1 can be written in the form below, where a is a scalar, v is in ℝk , and A is a k × k lower triangular matrix. See the Study Guide for help with induction.]

Step-by-Step Solution:
Step 1 of 3

Solution 24E

From the product above, =

So, the result is true for

By the principle of mathematical induction the result is true for all

For all matrices A and B of the type as specified of order n, .

According to the invertible matrix theorem matrix A is invertible and B is its inverse.

Step 2 of 3

Step 3 of 3

##### ISBN: 9780321982384

Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since the solution to 24E from 2.4 chapter was answered, more than 505 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 24E from chapter: 2.4 was answered by , our top Math solution expert on 07/20/17, 03:54AM. The answer to “Use partitioned matrices to prove by induction that for n = 2, 3,…,the n × n matrix A shown below is invertible and B is its inverse. For the induction step, assume A and B are (k + 1) × (k + 1) matrices, and partition A and B in a form similar to that displayed in Exercise 23.Exercise 23:Use partitioned matrices to prove by induction that the product of two lower triangular matrices is also lower triangular. [Hint: A(k + 1) × (k + 1) matrix A1 can be written in the form below, where a is a scalar, v is in ?k , and A is a k × k lower triangular matrix. See the Study Guide for help with induction.]” is broken down into a number of easy to follow steps, and 123 words. This full solution covers the following key subjects: Matrices, induction, Matrix, triangular, Lower. This expansive textbook survival guide covers 65 chapters, and 1898 solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5.

#### Related chapters

Unlock Textbook Solution