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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 5: Q7E (page 263)

All nonzero symmetric matrices are invertible.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Definition of a symmetric matrix.

Suppose A is a $n 脳 n$ matrix.

Then the matrix is said to be symmetric matrix, if $A^{T} = A$ .

A matrix is said to be invertible, if the determinant of the matrix is non-zero.

02

Check whether the given statement is a true or false.

The given statement is all nonzero symmetric matrices are invertible.

For example.

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

Then,

$A^{T} = {[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]}^{T} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] = A A^{T} = {[\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}]}^{T} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] = A$

So, A is a symmetric matrix, but it is not invertible, because det(A)=0.

This means,every symmetric matrix cannot be invertible.

Then the given statement is false.

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Most popular questions from this chapter

Consider the orthonormal vectors ${\overset{鈫�}{u}}_{1}, {\overset{鈫�}{u}}_{2}, {\overset{鈫�}{u}}_{3}, {\overset{鈫�}{u}}_{4}, {\overset{鈫�}{u}}_{5}$ in ${鈩�}^{10}$ . Find the length of the vector $\overset{鈫�}{x} = 7 {\overset{鈫�}{u}}_{1} - 3 {\overset{鈫�}{u}}_{2} + 2 {\overset{鈫�}{u}}_{3} + {\overset{鈫�}{u}}_{4} - {\overset{鈫�}{u}}_{5}$ .

TRUE OR FALSE?If matrix A is orthogonal, then the matrix $A^{T}$ must be orthogonal as well.

Consider the vectors

$\overset{r}{u} = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ . \\ 1 \end{matrix}] a n d \overset{r}{v} = [\begin{matrix} 1 \\ 1 \\ . \\ . \\ . \\ 1 \end{matrix}] in R^{n}$

a.For n =2,3,4 , find the angle $胃$ between role="math" localid="1659432008110" $\overset{鈯�}{u}$ and $\overset{鈯�}{v}$ . For $n = 2$ and 3, represent the vectors graphically.

b.Find the limit of $胃$ as napproaches infinity.

Let n be an even integer.In both parts of this problem,let Vbe the subspace of all vector $\overset{鈫�}{x}$ in $R^{n}$
such that $x_{j + 2} = x_{j} + x_{j + 1} 鈭赌 j = 1, 2, . ., n 鈭� 2 .$ .Consider the basis $\overset{鈫�}{v}, \overset{鈫�}{w}$ of V with

$\overset{鈫�}{a} = [\begin{array}{l} 1 \\ a \\ a^{2} \\ . \\ . \\ . \\ a^{n 鈭� 1} \end{array}], \overset{鈫�}{b} = [\begin{array}{l} 1 \\ b \\ b^{2} \\ . \\ . \\ . \\ b^{n 鈭� 1} \end{array}]$

where $a = \frac{1 + \sqrt{5}}{2}$ and $b = \frac{1 鈭� \sqrt{5}}{2}$

a.Show that $\overset{鈫�}{a} 鈥�$ is orthogonal to $\overset{鈫�}{b}$

b.Explain why the matrix P of the orthogonal projection onto V is a Hankel matrix.

If the nxn matrices Aand Bare symmetric and Bis invertible, which of the matrices in Exercise 13 through 20 must be symmetric as well?AB.

See all solutions

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