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Chapter 7: Q43E (page 346)

For which values of constants a, b, and c are the matrices in Exercises 40 through 50 diagonalizable?

A=[11a1]

Short Answer

Expert verified

The matrix .a鈮�0 So, is diagonalizable.

Step by step solution

01

Characteristic equation 

Consider an n 脳n matrix A and a scalar 位. Then 位 is an eigenvalue5 of A.

det(A-位濒)=0

This is called the characteristic equation (or the secular equation) of matrix A.

02

Solution of the problem

We solve:

detA-位濒=01-位1a1-位=01-位2-a=0位2-2位+1-a=0位1,2=2卤4-41-a2=1卤a

For a=0, we only have one eigenvalue, so A is not diagonalizable. On the other hand, for a鈮�0, we have two distinct eigenvalues, so A is diagonalizable.

Here, the final result is a鈮�0.

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Most popular questions from this chapter

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?

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 through 20,find all (real) eigenvalues. Then find a basis of each eigenspaces ,and diagonalize A, if you can. Do not use technology.

[6327]

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