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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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