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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: Q7E (page 289)

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.
7. $[\begin{matrix} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 & 5 \end{matrix}]$

Short Answer

Expert verified

Therefore, the determinant of given matrix is given by,

$d e t A = 1$

Step by step solution

01

Definition

Gaussian elimination method is used to solve a system of linear equations.

Gaussian elimination provides a relatively efficient way of constructing the inverse to a matrix.

Interchanging two rows. Multiplying a row by a constant (any constant which is not zero).

02

Given

Given Matrix,

$A = [\begin{matrix} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 1 & 2 & 3 & 4 \\ 1 & 2 & 3 & 4 & 5 \end{matrix}]$

03

To find determinant by using Gaussian Eliminations

We switch the first and fifth row, and then the second and fourth row. We get

We get,

$A = [\begin{matrix} 1 & 2 & 3 & 4 & 5 \\ 0 & 1 & 2 & 3 & 4 \\ 0 & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]$

We had two row interchanges, so

$d e t A = {(- 1)}^{2} 路 1 路 1 路 1 路 1 路 1 d e t A = 1$

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Most popular questions from this chapter

Consider a $4 脳 4$ matrix A with rows ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}, {\overset{鈫�}{v}}_{4}$ . If, find the determinants in Exercises 11 through16.

12. localid="1659509477853" $d e t [\begin{matrix} {\overset{鈫�}{v}}_{1} \\ {\overset{鈫�}{v}}_{2} \\ {\overset{鈫�}{v}}_{3} \\ {\overset{鈫�}{v}}_{4} \end{matrix}]$

Find the determinants of the linear transformations in Exercises 17 through 28.

19. $T (f) = 2 f + 3 f' from P_{2} to P_{2}$

If is any noninvertible square matrix, then $d e t A = d e t (r r e f A)$ .

$d e t (4 A) = 4 d e t A$ for all $4 脳 4$ matricesA.

Find the determinants of the linear transformations in Exercises 17 through 28.

21. $T (f (t)) = f (- t) f r o m P_{3} t o P_{3}$

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