Question
Let A =
& I =
and A-1= 
[A2+CA+dI], find ordered pair (c,d) ?
- (6, 11)
- (-6, -11)
- (-6 , 11)
- (6, -11)
Hint:
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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