Question

Determine the algebraic and geometric multiplicities of the eigenvalues for the following matrix. B = 13 71 has characteristic equation (3-1)(6 - 1) = 0 LO 6] First determine the eigenvalues, order them from smallest to greatest: 11 = 12 = Now determine the algebraic and geometric multiplicities for each eigenvalue above. You can do this with direct computation or using any of the theorems discussed in class to avoid computation. ab(11) = YB(11) = ab(12) = YB(12) = We conclude that the matrix (enter "is" or "is not") diagonalizable.

Answer 1

Given that B=13 7 ) has characteristic equation (3-2)(6-)=0. Therefore, the eigenvalues are 3,6 ..so, x= 3, d2=6 We know that a The algebric multiplicity of an eigen value of B is the number of times & appears as a roof of charecteristic equation." and " the geometric multiplicity of an eigenvalued of B is the dimension of the eigenspace associated with the eigenvalued." Therefore, dB (d.)=1, dB (dz)=1 clearly, the dimension of the eigenspace of both the eigenvalues are 1. so, Vold,)=1, VB (dz) =1. Again we know that a ff B is a square materike of ordern and B has n distinct eigenvalues, then B is diagonalizable.". 1 Here, B has two distinct eigenvalues and B is of order 2 . so, the matrix B is diagonalizable .