if A=[0  1  1]   and B = 1/2 [b+c    c-a   b-a]
        [1  0  1]                       [c-b     c+a   a-b]
       [1  1  0]                        [b-c     a-c    a+b]    
  show that ABA-1 is a diagonal matrix .

Dear student
A=011101110A=0-10-1+11-0=1+1=20Let Cij be the cofactor of aij in A=aij.Then,C11=-11+10-1=-1C12=-11+20-1=1C13=-11+31-0=1C21=-12+10-1=1C22=-12+20-1=-1C23=-12+30-1=1C31=-13+11-0=1C32=-13+20-1=1C33=-13+30-1=-1adj(A)=-1111-1111-1T=-1111-1111-1So, A-1=1Aadj(A)=12-1111-1111-1Now,Consider,ABA-1=14011101110b+cc-ab-ac-bc+aa-bb-ca-ca+b-1111-1111-1=14c-b+b-cc+a+a-ca-b+a+bb+c+b-cc-a+a-cb-a+a+bb+c+c-bc-a+c+ab-a+a-b-1111-1111-1=1402a2a2b02b2c2c0-1111-1111-1=142a+2a-2a+2a2a-2a-2b+2b2b+2b2b-2b-2c+2c2c-2c2c+2c=144a0004b0004c=a000b000cwhich is a diagonal matrix
Regards

  • 0
What are you looking for?