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Chapter 7: Q1E (page 382)

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

Short Answer

Expert verified

True,detA=0

Step by step solution

01

Define eigenvalue:

Eigenvalues are a set of specialized scales associated with a system of linear equations. The corresponding eigenvalue, often denoted by 位.

02

Explanation for the det A = 0:

If detA鈮�0i.e. the matrix is invertible, then the linear transformation whose matrix in a basis is A is an isomorphism, thus a monomorphism, Therefore, kerA=0 So, 0 is not an eigenvalue of A. Hence, if 0 is an eigenvalue of A then,detA=0

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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

8((110022003)

For m鈮�n, find the dimension of the space of alln脳nmatricesfor which all the vectorse鈬赌1,....,e鈬赌mare eigenvectors.

For an arbitrary positive integer n, give a 2n脳2n matrix A without real eigenvalues.

For which2脳2matrices A does there exist a invertible matrix M Such that AS=SD,whereD=[2003] Give your answer in terms of eigenvalues of A.

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.

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