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Chapter 7: Q19E (page 336)

True or false? If the determinant of a 2 脳 2 matrix A is negative, then A has two distinct real eigenvalues.

Short Answer

Expert verified

We have two distinct eigenvalues. The given statement is true.

Step by step solution

01

Eigenvalues

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled.

02

We have to tell true or false of the statement above

As we clearly see,

A=abcdad-bc<0鈬�ad<bc

detA-位l=0a-位bcd-位=0a-位d-位-bc=0位1,2=a+d卤a-d2-4ad-bc2

Since, ad - bc , then the expression under the root sign is always positive, so these are two distinct numbers.

Thus, we have two distinct eigenvalues.

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Most popular questions from this chapter

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.

find an eigenbasis for the given matrice and diagonalize:

A=114[13-2-3-210-6-3-65]

representing the orthogonal projection onto a plane E.

Consider the matrixA=[2034] Show that 2 and 4 are eigenvalues ofand find all corresponding eigenvectors. Find an eigen basis for Aand thus diagonalizeA.

Find all 4x4matrices for whiche鈬赌2is an Eigen-vector.

Give an example of a 4脳4matrix A without real eigenvalues.
See all solutions

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