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Linear Algebra With Applications

Otto Bretscher

$Math Studyset Vaia Explanations$ Math

5th Edition · 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

Determine whether the statements that follow are true or false, and justify your answer.

17: Rank $| \begin{matrix} 2 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{matrix} | = 2$

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$A = [\begin{matrix} 1 & 1 \\ 10^{- 10} & 0 \\ 0 & 10^{- 10} \end{matrix}] and \overset{⇀}{b} = [\begin{matrix} 1 \\ 10^{- 10} \\ 10^{- 10} \end{matrix}]$ .

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Determine whether the statements that follow are true or false, and justify your answer.

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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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Most popular questions from this chapter

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