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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: Q33E (page 309)

If Ais any symmetric matrix, then detA = 1 or det A = -1.

Short Answer

Expert verified

Therefore, det A = 0 the given statement is not satisfied.

Step by step solution

01

Matrix Definition

Matrix is a set of numbers arranged in rowsand columns so as to form a rectangular array.

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

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

02

To check whether the given condition is true or false

For example,

$A = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$

$d e t A = 1 - 1$ .

is a symmetric matrix, yet $d e t A = 0$ .

Therefore, the given statement is false.

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Most popular questions from this chapter

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

24. $T (z) = (2 + 3 i) z f r o m 鈩� t o 鈩�$

If Ais any n x n matrix, then $d e t (A A^{T}) = d e t (A^{T} A)$ .

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.

Consider the function $F (A) = F [\overset{鈫�}{v} \overset{鈫�}{w}] = \overset{鈫�}{v} 脳 \overset{鈫�}{w}$ from ${鈩�}^{2 脳 2}$ to $鈩�$ , the dot product of the column vectors of A.
a. Is Flinear in both columns of A? See Example 6.
b. Is F linear in both rows of A?
c. Is Falternating on the columns of A? See Example 4.

The matrix $[\begin{matrix} k^{2} & 1 & 4 \\ k & - 1 & - 2 \\ 1 & 1 & 1 \end{matrix}]$ is invertible for all positive constantsk.

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