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Chapter 7: Q7-18E (page 383)

TRUE OR FALSE

18. If A and B are nxn matrices, if 伪is an eigenvalue of A, and if 尾is an eigenvalue of B, then 伪尾 must be an eigenvalue of AB.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Define eigenvalues

The scalar values that are associated with the vectors of the linear equations in the matrix are called eigenvalues.

Ax鈫�=位x鈫�, herex鈫�is eigenvector and位 is the eigenvalue.

02

Find the eigenvalues by assuming the matrix

Consider the matrix,

A=2100andB=0003

Hence,伪=2,尾=3are the eigenvalues of the matrices A and B respectively.

Multiply the matrices,

AB=0300

Thus,伪尾=6and it is not an eigenvalue of AB.

Therefore, the given statement is false, since伪尾is not an eigenvalue of AB.

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Most popular questions from this chapter

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

15.(-101-301403)

Consider an upper triangular n脳nmatrix Awithaij鈮�0fori=1,2,鈥�,mandaij=0fori=m+1,鈥�n. Find the algebraic multiplicity of the eigenvalueof. Without using Theorem 7.3.6, what can you say about the geometric multiplicity?

If 0 is an eigenvalue of a matrix A, then det A = 0.

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.

[2-2001-100003-4002-3]

For a given eigenvalue, find a basis of the associated eigenspace. Use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable. For each of the matrices A in Exercises 1 through 20, find all (real) eigenvalues. Then find a basis of each eigenspace, and diagonalize A, if you can. Do not use technology

0-112
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