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Chapter 7: Q46E (page 346)

For which values of constants a,b,c are the matrix diagonalizable?

[1ab02c001]

Short Answer

Expert verified

The matrix is A diagonalizable for the value ofa,b,c鈭�鈩�,b=ac

Step by step solution

01

Calculating the Characteristic Equation of the matrix .

Matrix A is diagonalizable only if the geometric multiplicities (gemu) of the eigenvalues add up to n.

The given matrix is1ab02c001

This matrix has an upper triangular shape. As a result, the entries on its main diagonal are its eigenvalues.

They are 位1=1,位2=1,位3=2

Since almu (2) =1 then gemu (2) = 1. Therefore , we need to calculate only the gemu (1),

(A-l)x=00ab01c000x1x2x3=000ax2+bx3=0,x2+cx3=0ax2+bx3=0,ax2+acx3=0

02

Analyzing whether the matrix is diagonalizable.

For b鈮�ac, we get x2=x3= 0. So,

E1=100

Hence , gemu (1) = 1 Therefore the matrix is not diagonalizable.

For b = ac , we get ax2+acx3=0. So

E1=100,0c-1

Hence gemu (1) = 2. Therefore the matrix A is diagonalizable.

Thus the matrix A is diagonalizable for the value ofa,b,c鈭�,b=ac

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