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Chapter 7: Q48E (page 384)

If a matrix A has k distinct eigenvalues, then (A)鈮�K

Short Answer

Expert verified

False, for a matrix A has k distinct eigenvalues, thenA鈮�K

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 a matrix A has k distinct eigenvalues:

Consider an example,

A=0011

This matrix is lower-triangular matrix, so its eigenvalues are the entries on its main diagonal, which are 位1=0,位2=1meaning k = 2. But the rows of this matrix are linearly dependent, so rank A = 1 < K. Hence it is false.

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