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?

Answered: Form the converse, inverse, and…

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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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 $p \to q$

Converse Statement is $q \to p$

"If A is diagonalizable, then an n x n matrix A has n distinct real eigenvalues"

Inverse statement is $~ p \to ~ q$

"If an n x n matrix A has not n distinct real eigenvalues, then A is not diagonalizable."

Contrapositive statement is $~ q \to ~ p$

"If A is not diagonalizable, then an n x n matrix A has not n distinct real eigenvalues"

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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?