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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: Q64E (page 293)

Show that fewer thane.n! Algebraic operations (additions and multiplications) are required to compute the determinant of a $n 脳 n$ matrix by Laplace expansion. Hint: Let $L_{n}$ is the number of operations required to compute the determinant of a "general" $n 脳 n$ matrix by Laplace expansion. Find a formula expressing $L_{n}$ in terms of $L_{n - 1}$ . Use this formula to show, by induction (see Appendix B.1), that $\frac{L_{n}}{n!} = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + L + \frac{1}{(n - 1)!} - \frac{1}{n!}$ Use the Taylor series of $e^{x}, e^{x} = {鈭�}_{n - 0}^{鈭�} \frac{x^{n}}{n!}$ , to show that the right-hand side of this equation is less than e.

Short Answer

Expert verified

Therefore, the expression of $L_{n}$ is given by, $L_{n} < e n!, 獗� n 鈭� 鈩�$ .

Step by step solution

01

Definition

A determinant is a unique number associated with asquare matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

02

To find the formula expressing in Ln terms

Let $L_{n}$ be the number of algebraic operations required to calculate the determinant of a $n 脳 n$ matrix using the Laplace expansion. We have:

$L_{1} = 0, L_{n} = n (L_{n - 1} + 2) - 1$ ,This gives us:

$\frac{L_{n}}{n!} = \frac{L_{n - 1}}{(n - 1)!} + \frac{2}{(n - 1)!} - \frac{1}{n!}$ This, in recursion, gives us:

role="math" localid="1660718145433" $\frac{L_{n}}{n!} = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + . . . + \frac{1}{(n - 1)!} - \frac{1}{n!} < {鈭�}_{n - 0}^{鈭�} \frac{1}{n!}$

$\frac{L_{n}}{n!} = E$ Which, finally, gives us $L_{n} < e n!, 獗� n 鈭� 鈩�$ .

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Most popular questions from this chapter

Use Gaussian elimination to find the determinant of the matrices A in Exercises 1 through 10.

4. $[\begin{matrix} 1 & - 1 & 2 & - 2 \\ - 1 & 2 & 1 & 6 \\ 2 & 1 & 14 & 10 \\ - 2 & 6 & 10 & 33 \end{matrix}]$

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 det(A) = 8, find the determinants in Exercises 11 through16.

13. $d e t [\begin{matrix} {\overset{鈫�}{v}}_{1} \\ {\overset{鈫�}{v}}_{2} \\ {\overset{鈫�}{v}}_{3} \\ {\overset{鈫�}{v}}_{4} \end{matrix}]$

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

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

Vandermonde determinants (introduced by Alexandre-Th茅ophile Vandermonde). Consider distinct real numbers $a_{0}, a_{1}, . . . . ., a_{n .}$ . We define $(n + 1) 脳 (n + 1)$ the matrix

$A = [\begin{matrix} 1 & 1 & . . . . 1 \\ a_{0} & a_{1} & . . . . a_{n} \\ a_{0}^{2} & a_{1}^{2} & . . . . a_{1}^{2} \\ a_{0}^{n} & a_{1}^{n} & . . . . a_{n}^{n} \end{matrix}]$

Vandermonde showed that

$d e t (A) = \underset{i > j}{鈭�} (a_{i} - a_{j})$

the product of all differences $(a_{i} - a_{j})$ , where exceeds j.
a. Verify this formula in the case of $n = 1$ .
b. Suppose the Vandermonde formula holds for $n = 1$ . You are asked to demonstrate it for n. Consider the function

$f (t) = \det [\begin{matrix} 1 & 1 & . . . & 1 & 1 \\ a_{0} & a_{1} & . . . & a_{n - 1} & t \\ a_{0}^{2} & a_{1}^{2} & . . . & a_{n - 1} & t^{2} \\ 鈰� & 鈰� & . . . & 鈰� & 鈰� \\ a_{0}^{n} & a_{1}^{n} & . . . & a_{n - 1}^{n} & t^{n} \end{matrix}]$

Explain why f(t) is a polynomial of $n^{t h}$ degree. Find the coefficient k of $t^{n}$ using Vandermonde's formula for $a_{0}, . . ., a_{n - 1}$ . Explain why

role="math" localid="1659522435181" $f (a_{0}) = f (a_{1}) = . . . = f (a_{n - 1}) = 0$

Conclude that

$f (t) = k (t - a_{0}) (t - a_{1}) . . . (t - a_{n - 1})$

for the scalar k you found above. Substitute $t = a_{n}$ to demonstrate Vandermonde's formula.

If an $n 脳 n$ matrixAis invertible, then there must be an $(n - 1) 脳 (n - 1)$ sub matrix of(obtained by deleting a row and a column of) that is invertible as well.

See all solutions

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