Q33E For n聽distinct scalars a1,a2,..... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q33E (page 393)

For ndistinct scalars $a_{1}, a_{2}, . . ., a_{n}$ , find
role="math" localid="1659609991385" $d e t [\begin{matrix} a_{1} & a_{2} & 鈥� & a_{n} \\ a_{1}^{2} & a_{2}^{2} & 鈥� & a_{n}^{2} \\ 鈰� & 鈰� & 鈰� \\ a_{1}^{n} & a_{2}^{n} & 鈥� & a_{n}^{n} \end{matrix}]$

Short Answer

Expert verified

Therefore, the determinant is given by,

$\det [\begin{matrix} a_{1} & a_{2} & 鈥� & a_{n} \\ a_{1}^{2} & a_{2}^{2} & 鈥� & a_{n}^{2} \\ 鈰� & 鈰� & 鈰� \\ a_{1}^{n} & a_{2}^{n} & 鈥� & a_{n}^{n} \end{matrix}] = ({鈭�}_{k = 1}^{n} (a_{i} - a_{j}))$ .

Step by step solution

By using Vandermonde’s Formula.

$For, we have a matrix A = [\begin{matrix} 1 & 1 \\ a_{0} & a_{1} \end{matrix}] Using Vandermonde' s formula, we have \underset{i > j}{鈭� (a_{i} - a_{j})} = a_{1} - a_{0} = \det A .$

To find the determinant.

Using Vandermonde's formula, we compute

$\det [\begin{matrix} a_{1} & a_{2} & 鈥� & a_{n} \\ a_{1}^{2} & a_{2}^{2} & 鈥� & a_{n}^{2} \\ 鈰� & 鈰� & 鈰� \\ a_{1}^{n} & a_{2}^{n} & 鈥� & a_{n}^{n} \end{matrix}] = ({鈭�}_{k = 1}^{n} a_{k}) \det [\begin{matrix} 1 & 1 & 鈥� & 1 \\ a_{1} & a_{2} & 鈥� & a_{n}^{} \\ 鈰� & 鈰� & 鈰� \\ a_{1}^{n - 1} & a_{2}^{n - 1} & 鈥� & a_{n}^{n - 1} \end{matrix}] \det [\begin{matrix} a_{1} & a_{2} & 鈥� & a_{n} \\ a_{1}^{2} & a_{2}^{2} & 鈥� & a_{n}^{2} \\ 鈰� & 鈰� & 鈰� \\ a_{1}^{n} & a_{2}^{n} & 鈥� & a_{n}^{n} \end{matrix}] = ({鈭�}_{k = 1}^{n} a_{k}) ({鈭�}_{\underset{i > j}{i, j = 1}}^{n} (a_{i} - a_{j})) Therefore, the determinant is given by, \det [\begin{matrix} a_{1} & a_{2} & 鈥� & a_{n} \\ a_{1}^{2} & a_{2}^{2} & 鈥� & a_{n}^{2} \\ 鈰� & 鈰� & 鈰� \\ a_{1}^{n} & a_{2}^{n} & 鈥� & a_{n}^{n} \end{matrix}] = ({鈭�}_{k = 1}^{n} a_{k}) ({鈭�}_{\underset{i > j}{i, j = 1}}^{n} (a_{i} - a_{j}))$

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91影视

For ndistinct scalars $a_{1}, a_{2}, . . ., a_{n}$ , find
role="math" localid="1659609991385" $d e t [\begin{matrix} a_{1} & a_{2} & 鈥� & a_{n} \\ a_{1}^{2} & a_{2}^{2} & 鈥� & a_{n}^{2} \\ 鈰� & 鈰� & 鈰� \\ a_{1}^{n} & a_{2}^{n} & 鈥� & a_{n}^{n} \end{matrix}]$

Short Answer

Step by step solution

By using Vandermonde’s Formula.

To find the determinant.

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91影视

For ndistinct scalars a1,a2,...,an, findrole="math" localid="1659609991385" det[a1a2鈥�ana12a22鈥�an2鈰�鈰�鈰�a1na2n鈥�ann]

Short Answer

Step by step solution

By using Vandermonde’s Formula.

To find the determinant.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

For ndistinct scalars $a_{1}, a_{2}, . . ., a_{n}$ , find
role="math" localid="1659609991385" $d e t [\begin{matrix} a_{1} & a_{2} & 鈥� & a_{n} \\ a_{1}^{2} & a_{2}^{2} & 鈥� & a_{n}^{2} \\ 鈰� & 鈰� & 鈰� \\ a_{1}^{n} & a_{2}^{n} & 鈥� & a_{n}^{n} \end{matrix}]$