Q47E For which values of constants a,... [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: Q47E (page 346)

For which values of constants a,b,c are the matrix diagonalizable?
$[\begin{matrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}]$

Short Answer

Expert verified

The matrix A is diagonalizable for the values of a = b = c = 0

Step by step solution

Calculating the value of of the matrix .

Consider an $n 脳 n$ matrix A and a scalar $位$ . Then $位$ is the eigenvalue of A if $\det (A - 位濒) = 0$ . This is called Characteristic equation of matrix .

The given matrix is $[\begin{matrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}]$

This matrix has an upper triangular shape. As a result, the entries on its main diagonal are its eigenvalues.

They are $位_{1} = 1, 位_{2} = 1, 位_{3} = 1$

Therefore , we need to calculate for $位 = 1$ ,

$(A - l) x = 0 |\begin{matrix} 0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{matrix}| |\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \end{matrix}| = \begin{matrix} 0 \\ 0 \\ 0 \end{matrix}$

Analyzing whether the matrix is diagonalizable.

For $c 鈮� 0$ , we get $x_{3} = 0$ . So, $E_{1}$ will be at most two-dimensional. For $c = 0, a 鈮� 0 鈮� b$ ,we will get ${ax}_{2} + {bx}_{3} = 0$

$E_{1} = \begin{matrix} 1 & 0 \\ 0 & b \\ 0 & - a \end{matrix}$ Which is also a two-dimensional. So the matrix is diagonalizable only if

Finally we can conclude that A is diagonalizable for all a = b = c = 0

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91影视

For which values of constants a,b,c are the matrix diagonalizable?
$[\begin{matrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}]$

Short Answer

Step by step solution

Calculating the value of of the matrix .

Analyzing whether the matrix is diagonalizable.

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91影视

For which values of constants a,b,c are the matrix diagonalizable?[1ab01c001]

Short Answer

Step by step solution

Calculating the value of of the matrix .

Analyzing whether the matrix is diagonalizable.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Pure Maths

Geometry

Mechanics Maths

Statistics

Study anywhere. Anytime. Across all devices.

For which values of constants a,b,c are the matrix diagonalizable?
$[\begin{matrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \end{matrix}]$