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Chapter 7: Q7-17E (page 383)

TRUE OR FALSE

17. If 1 is the only eigenvalue of a n脳n matrixA, then A must be ln.

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 given statement is true or false:

Consider the matrix,

A=1101

Since, the matrix is upper triangular matrix, its eigenvalues are the entries on its main diagonal, which are位1,2=1.

Hence, it is clear that A鈮�I2.

Therefore, 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.

[51-521082-7]

Consider a 4 脳 4 matrixA=|BC0D|where B, C, and D are 2 脳 2 matrices. What is the relationship among the eigenvalues of A, B, C, and D?

Consider the matrix A=|abba|where aand bare arbitrary constants. Find all eigenvalues of A.

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

For a given eigenvalue, find a basis of the associated eigensspace .use the geometric multiplicities of the eigenvalues to determine whether a matrix is diagonalizable.

For each of the matrices A in Exercise1 through20,find all (real)eigenvalues.Then find a basis of each eigenspaces,and diagonalize A, if you can. Do not use technology.

[1111]
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