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Question

Let A =open square brackets table row 1 0 0 row 0 1 1 row 0 cell negative 2 end cell 4 end table close square brackets& I =open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets and A-1= fraction numerator 1 over denominator 6 end fraction [A2+CA+dI], find ordered pair (c,d) ?

  1. (6, 11)    
  2. (-6, -11)    
  3. (-6 , 11)    
  4. (6, -11)    

hintHint:

use the characteristic equation and find out the determinant equation and equate it with 0

The correct answer is: (-6 , 11)



    {-6,11}
    Using the characteristic equation of square matrices, we can say that
    |A- λI| = 0
    This gives us λ3 -6 λ2 +11 λ-6 =0
    Or, A3 -6 A2 +11 A-6I =0
    As per the given problem, 6A -1 = A2+cA+dI
    Multiplying both sides by A, we get
    6I = A3 +cA2 +dA
    A3 +c A2 +d A-6I=0
    On comparing, we get c,d = {-6,11}

    the roots of the characteristic equation give us the eigenvalues of the matrix.

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