Determine whether A is similar B .If A∼B, give an invertible matrix P such that P−1AP=B A=[110011001],B=[110010001]

Question

Determine whether A is similar B .If A∼B, give an invertible matrix P such that P−1AP=B  A=[110011001],B=[110010001]

cyhuddwyr9 · Accepted Answer

Step 1 The given matrices are A=[110011001] and B=[110010001] Here, it is observed that the matrices A and B are diagonal matrix. Therefore, the eigenvalues are the entries on the main diagonal values. Hence, the eigenvalues of both A, B are λ1=1,λ2=1and λ3=1 Step 2 Obtain the eigenvector corresponding to the eigenvalue λ=1 by solving the system (A-I)X=0. Let X=[x1x2x3] Then, ([110011001]−[110010001])[x1x2x3]=0 [010001000][x1x2x3]=0 The system of equations corresponding to the above system is x1=x1(free variable) x2=0 x3=0 If x1=1 then v1=[100] Step 3 Obtain the eigenvector corresponding to the eigenvalue λ=1 by solving the system (B-I)X=0. Then ([110010001]−[100010001])[x1x2x3]=0 [010000000][x1x2x3]=0 The system of equations corresponding to the above system is x1=x1(free variable) x2=0 x3=x3(free variable) If x1=1,x3=0 then v1=[100] Choose the arbitrary value x1=0,x3=1 Then the eigenvector is v2=[001] It is observed that the eigenspace corresponding to λ=1 of a matrix B has dimension 2 and the eigenspace corresponding to λ=1 of a matrix A has dimension 1. Hence, the matrices are not similar.

Jeffrey Jordon · Accepted Answer

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Determine whether A is similar B .If A sim B, give an invertible matrix P such that P^{-1}AP = B A=begin{bmatrix}1 & 1&0 0&1&10&0&1 end{bmatrix} , B=begin{bmatrix}1&1&0 0&1&00&0&1 end{bmatrix}

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