Q7.5-47E 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: Q7.5-47E (page 375)

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

Short Answer

Expert verified

The value of the $a 鈮� 0$ .

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.

Solving the Equation

Here the values of the real constantare the matrices Diagonalizable

$A = (\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & a \\ 0 & 1 & 0 \end{matrix}) \det (A - 位 I) = 0 (\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & a \\ 0 & 1 & 0 \end{matrix}) - 位 (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) = 0 (\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & a \\ 0 & 1 & 0 \end{matrix}) - (\begin{matrix} 位 & 0 & 0 \\ 0 & 位 & 0 \\ 0 & 0 & 位 \end{matrix}) = 0 (\begin{matrix} - 位 & 0 & 0 \\ 1 & - 位 & a \\ 0 & 1 & - 位 \end{matrix}) = 0$

Now,

$\begin{array}{rcl} |\begin{matrix} - 位 & 0 & 0 \\ 1 & - 位 & a \\ 0 & 1 & - 位 \end{matrix}| & = & 0 \\ - 位 (位^{2} - a) - 0 (- 位 - 0) + (1 + 0) & = & 0 \\ - 位 (位^{2} - a) & = & 0 \\ 位 & = & 0, 位^{2} - a = 0 \\ 位_{1} & = & 0, 位_{2, 3} = 卤 \sqrt{a} \end{array}$

If $a 鈮� 0$ , three distinct eigenvalues for $a = 0$ matrix.

Therefore, the value of $a 鈮� 0$ .

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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?
47. $[\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & a \\ 0 & 1 & 0 \end{matrix}]$

Short Answer

Step by step solution

Define Matrix and Diagonalizable

Solving the 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?47.[00010a010]

Short Answer

Step by step solution

Define Matrix and Diagonalizable

Solving the Equation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Decision Maths

Logic and Functions

Mechanics Maths

Theoretical and Mathematical Physics

Discrete Mathematics

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?
47. $[\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & a \\ 0 & 1 & 0 \end{matrix}]$