Form the converse, inverse, and contrapositive of the statement: "If an n x n matrix A has n distinct real eigenvalues, then A is diagonalizable." Which of them are true?
Form the converse, inverse, and contrapositive of the statement: "If an n x n matrix A has n distinct real eigenvalues, then A is diagonalizable." Which of them are true?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Form the converse, inverse, and contrapositive of the statement:
"If an n x n matrix A has n distinct real eigenvalues, then A is diagonalizable."
Which of them are true?
Expert Solution
Step 1
Given statement is
"If an n x n matrix A has n distinct real eigenvalues, then A is diagonalizable."
p=n x n matrix has n distinct real eigenvalues
q= A is diagonalizable
Given statement is
Converse Statement is
"If A is diagonalizable, then an n x n matrix A has n distinct real eigenvalues"
Inverse statement is
"If an n x n matrix A has not n distinct real eigenvalues, then A is not diagonalizable."
Contrapositive statement is
"If A is not diagonalizable, then an n x n matrix A has not n distinct real eigenvalues"
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