Q7.5-50E For which values of the real con... [FREE SOLUTION]

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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 7: Q7.5-50E (page 375)

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

Short Answer

Expert verified

The Value of the Matrix $E_{0} = s p a n \{[\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}]\}$

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.

Calculation the Equation

Here the given matrix values is:

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

Using the formula

$\det (A - 位 I) = 0$

Substituting the values in above formula

$[\begin{matrix} - a & a & - a \\ - a - 1 & a + 1 & - a - 1 \\ 0 & 0 & 0 \end{matrix}] - 位 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] = 0 [\begin{matrix} - a & a & - a \\ - a - 1 & a + 1 & - a - 1 \\ 0 & 0 & 0 \end{matrix}] - [\begin{matrix} 位 & 0 & 0 \\ 0 & 位 & 0 \\ 0 & 0 & 位 \end{matrix}] = 0 [\begin{matrix} - a - 位 & a & - a \\ - a - 1 & a + 1 - 位 & - a - 1 \\ 0 & 0 & - 位 \end{matrix}] = 0 |\begin{matrix} - a - 位 & a & - a \\ - a - 1 & a + 1 - 位 & - a - 1 \\ 0 & 0 & - 位 \end{matrix}| = 0$

Here,

$- a - 位 ((a + 1 - 位) (- 位)) - a ((- a - 1) (- 位)) - a (0) = 0 - a - 位 (- a 位 - 位 + 位^{2}) - a (a 位 + 位)) = 0 - a^{2} 位 + a 位 - a 位^{2} + a 位^{2} + 位^{2} - 位^{3} - a^{2} 位 - a 位 = 0 - 位^{2} - 位^{3} = 0 - 位^{2} (1 - 位) = 0 位_{1, 2} = 0, 位_{3} = 1$

Solving the Value

For any value $a 鈭� 鈩�$

$E_{0} = s p a n \{[\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 1 \end{matrix}]\}$

The algebraic and geometric will be multiple of eigenvalues, thus A is diagonalisable for all $a 鈭� 鈩�$ .

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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?
50. $[\begin{matrix} - a & a & - a \\ - a - 1 & a + 1 & - a - 1 \\ 0 & 0 & 0 \end{matrix}]$

Short Answer

Step by step solution

Define Matrix and Diagonalizable

Calculation the Equation

Solving the Value

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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?50.[-aa-a-a-1a+1-a-1000]

Short Answer

Step by step solution

Define Matrix and Diagonalizable

Calculation the Equation

Solving the Value

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Probability and Statistics

Decision Maths

Theoretical and Mathematical Physics

Discrete 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?
50. $[\begin{matrix} - a & a & - a \\ - a - 1 & a + 1 & - a - 1 \\ 0 & 0 & 0 \end{matrix}]$