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Chapter 7: Q7-16E (page 383)

TRUE OR FALSE

16. If A and B are two 3x3 matrices such that trA=trBanddetA=detB,then A and B must have the same eigenvalues.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Define matrix

A table of numbers and symbols arranged in the form of rows and columns is known as a matrix.

02

Consider the matrix and find the eigenvalues:

Eigenvalues are a unique set of scalar values connected to a set of linear equations that are most likely seen in matrix equations. The characteristic roots are another name for the eigenvectors. It is a non-zero vector that, after applying linear transformations, can only be altered by its scalar factor.

Let us take,

A=10002000andB=300000000

Hence,trA=trB=3and detA=detB=0, but the eigenvalues of matrix A are 1, 2 and 0, while the eigenvalues of matrix B are 0 and 0.0

Since, the eigenvalues of matrices A and B are different, the given statement is false.

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Most popular questions from this chapter

For each of the matrices in Exercises 1 through 13, find all real eigenvalues, with their algebraic multiplicities. Show your work. Do not use technology.

[010001100]

Show that similar matrices have the same eigenvalues. Hint:v鈫� Ifis an eigenvector ofS-1AS, thenrole="math" localid="1659529994406" Sv鈫�is an eigenvector of A.

Find an eigenbasis of given matrix and diagonalize it.

A=(1326)

If a 2 脳 2 matrix A has two distinct eigenvalues位1and位2, show that A is diagonalizable.

For which3脳3 matrices A does there exist a nonzero matrix M Such that AM=MD,where D=[200030004]Give your answer in terms of eigenvalues of A.
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