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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 1: Q35E (page 1)

If the determinants of all the principal submatrices of a symmetric 3X3matrixare negative, thenmust be negative definite.

Short Answer

Expert verified

The given statement is FALSE.

Step by step solution

Check whether the given statement is TRUE or FALSE

Consider

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

It is given that the determinants of the principle sub-matrices are negative.

However, $e_{2}^{T} {Ae}_{2} = 1 > 0$ implies that A is not a negative definite.

Final Answer

So, the given statement is FALSE.

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Most popular questions from this chapter

If A is a non-zero matrix of the form $[\begin{matrix} a & - b \\ b & a \end{matrix}]$ ,then the rank of A must be 2.

The momentum $\overset{鈫�}{P}$ of a system of n particles in space with masses $m_{1}, m_{2}, . ..., m_{n}$ and velocities $\overset{鈫�}{蠀_{1}}, \overset{鈫�}{蠀_{2}}, . ..., \overset{鈫�}{蠀_{n}}$ is defined as

$\overset{鈫�}{P} = m_{1} \overset{鈫�}{蠀_{1}} + m_{2} \overset{鈫�}{蠀_{2}} + . ... + m_{n} \overset{鈫�}{蠀_{n}}$

Now consider two elementary particles with velocities

$\overset{鈫�}{蠀_{1}} = [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}]$ and $\overset{鈫�}{蠀_{2}} = [\begin{matrix} 4 \\ 7 \\ 10 \end{matrix}]$

The particles collide. After the collision, their respective velocities are observed to be

$\overset{鈫�}{蝇_{1}} = [\begin{matrix} 4 \\ 7 \\ 4 \end{matrix}]$ and $\overset{鈫�}{蝇_{2}} = [\begin{matrix} 2 \\ 3 \\ 8 \end{matrix}]$

Assume that the momentum of the system is conserved throughout the collision. What does this experiment tell you about the masses of the two particles? See the accompanying figure.

In exercises, 1 through 10, find all solutions of the linear systems using elimination.Then check your solutions.

8. $|x + 2 y + 3 z = 0 4 x + 5 y + 6 z = 0 7 x + 8 y + 10 z = 0|$

Let A be a 4 脳 4 matrix, and let $\overset{鈫�}{b}$ and $\overset{鈫�}{c}$ be two vectors in ${鈩�}^{4}$ . We are told that the system $A \overset{鈫�}{x} = \overset{鈫�}{b}$ has a unique solution. What can you say about the number of solutions of the system $A \overset{鈫�}{x} = \overset{鈫�}{c}$ ?

Find the function $f (t)$ of the form $f (t) = a e^{3 t} + b e^{2 t}$ such that $f (0) = 1$ and $f' (0) = 4$ .

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91影视

If the determinants of all the principal submatrices of a symmetric 3X3matrixare negative, thenmust be negative definite.

Short Answer

Step by step solution

Check whether the given statement is TRUE or FALSE

Final Answer

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