Q7.5-48E For which values of the real con... [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 7: Q7.5-48E (page 375)

For which values of the real constant 鈥榓鈥� are the matricesin Exercises 45 through 50 diagonalizable over C?
48. $[\begin{matrix} 0 & 0 & a \\ 1 & 0 & 3 \\ 0 & 1 & 0 \end{matrix}]$

Short Answer

Expert verified

The value of $a 鈮� 卤 2$ .

Step by step solution

Define Matrix and Diagonalizable

A matrix is a rectangular array of numbers or symbols which are generally arranged in rows and columns.

Inlinear algebra, asquare matrix {\displaystyle A}is called diagonalizable or non-defective if it issimilar to adiagonal matrix.

Calculating the value of equation

Here the given values is:

$A = [\begin{matrix} 0 & 0 & a \\ 1 & 0 & 3 \\ 0 & 1 & 0 \end{matrix}] [\begin{matrix} 0 & 0 & a \\ 1 & 0 & 3 \\ 0 & 1 & 0 \end{matrix}] - 位 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = 0 [\begin{matrix} 0 & 0 & a \\ 1 & 0 & 3 \\ 0 & 1 & 0 \end{matrix}] - [\begin{matrix} 位 & 0 & 0 \\ 0 & 位 & 0 \\ 0 & 0 & 位 \end{matrix}] = 0 [\begin{matrix} - 位 & 0 & a \\ 1 & - 位 & 3 \\ 0 & 1 & - 位 \end{matrix}] = 0 |\begin{matrix} - 位 & 0 & a \\ 1 & - 位 & 3 \\ 0 & 1 & - 位 \end{matrix}| = 0 - 位 (位^{2} - 3) - 0 (- 位 - 0) + a (1 + 0) = 0 - 位^{3} + 3 位 + a = 0$

To consider the multiple root we compute

$\begin{array}{rcl} f' (位) & = & - 3 位^{2} + 3 \\ = & - 3 (位^{2} - 1^{2}) \\ = & - 3 (位 - 1) (位 + 1) \\ 位 & = & 卤 1 \\ 位 & = & + 1 鈬� - (1)^{2} + (3 脳 1) + a = 0 \\ - 1 + 3 + a & = & 0 \\ 2 + a & = & 0 \\ a & = & - 2 \\ 位 & = & - 1 鈬� - (- 1)^{3} + (3 脳 - 1) + a = 0 \\ 1 + (- 3) + a & = & 0 \\ - 2 + a & = & 0 \\ a & = & 2 \\ 鈭� a & = & 卤 2 \end{array}$

The candidates for the couple roots of $位 = 卤 1$ in which case we would have $a = 鈭� 2$ So, the A is been diagnosable at $a 鈮� 卤 2$ .

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91影视

For which values of the real constant 鈥榓鈥� are the matricesin Exercises 45 through 50 diagonalizable over C?
48. $[\begin{matrix} 0 & 0 & a \\ 1 & 0 & 3 \\ 0 & 1 & 0 \end{matrix}]$

Short Answer

Step by step solution

Define Matrix and Diagonalizable

Calculating the value of equation

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91影视

For which values of the real constant 鈥榓鈥� are the matricesin Exercises 45 through 50 diagonalizable over C?48. [00a103010]

Short Answer

Step by step solution

Define Matrix and Diagonalizable

Calculating the value of equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

Logic and Functions

Applied Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

For which values of the real constant 鈥榓鈥� are the matricesin Exercises 45 through 50 diagonalizable over C?
48. $[\begin{matrix} 0 & 0 & a \\ 1 & 0 & 3 \\ 0 & 1 & 0 \end{matrix}]$