Home > Matrix & Vector calculators > Inverse of matrix using Cayley Hamilton method calculator

 Solve any problem (step by step solutions) Input table (Matrix, Statistics)
Mode :
SolutionHelp
Solution
Problem: Cayley Hamilton inverse [[10,-9,-12],[7,-12,11],[-10,10,3]] [ Calculator, Method and examples ]

Solution:
Your problem -> Cayley Hamilton inverse [[10,-9,-12],[7,-12,11],[-10,10,3]]

To apply the Cayley-Hamilton theorem, we first determine the characteristic polynomial p(t) of the matrix A.
|A-tI|

=
 (10-t) -9 -12 7 (-12-t) 11 -10 10 (3-t)

=(10-t)((-12-t) × (3-t) - 11 × 10)-(-9)(7 × (3-t) - 11 × (-10))+(-12)(7 × 10 - (-12-t) × (-10))

=(10-t)((-36+9t+t^2)-110)+9((21-7t)-(-110))-12(70-(120+10t))

=(10-t)(-146+9t+t^2)+9(131-7t)-12(-50-10t)

= (-1460+236t+t^2-t^3)+(1179-63t)-(-600-120t)

=-t^3+t^2+293t+319

p(t)=-t^3+t^2+293t+319

The Cayley-Hamilton theorem yields that
O = p(A)=-A^3+A^2+293A+319I

Rearranging terms, we have
:. 319I = A^3-A^2-293A

:. 319I = A(A^2-A-293I)

:. A^-1 = 1/319(A^2-A-293I)

Now, first we find A^2-A-293I

A^2=A×A=
 10 -9 -12 7 -12 11 -10 10 3
×
 10 -9 -12 7 -12 11 -10 10 3

=
 10×10-9×7-12×-10 10×-9-9×-12-12×10 10×-12-9×11-12×3 7×10-12×7+11×-10 7×-9-12×-12+11×10 7×-12-12×11+11×3 -10×10+10×7+3×-10 -10×-9+10×-12+3×10 -10×-12+10×11+3×3

=
 100-63+120 -90+108-120 -120-99-36 70-84-110 -63+144+110 -84-132+33 -100+70-30 90-120+30 120+110+9

=
 157 -102 -255 -124 191 -183 -60 0 239

A^2 =
 10 -9 -12 7 -12 11 -10 10 3
2
=
 157 -102 -255 -124 191 -183 -60 0 239

A^2 - A =
 157 -102 -255 -124 191 -183 -60 0 239
-
 10 -9 -12 7 -12 11 -10 10 3
=
 157-10 -102+9 -255+12 -124-7 191+12 -183-11 -60+10 0-10 239-3
=
 147 -93 -243 -131 203 -194 -50 -10 236

293 × I = 293 ×
 1 0 0 0 1 0 0 0 1
=
 293 0 0 0 293 0 0 0 293

A^2 - A - 293 × I =
 147 -93 -243 -131 203 -194 -50 -10 236
-
 293 0 0 0 293 0 0 0 293
=
 147-293 -93+0 -243+0 -131+0 203-293 -194+0 -50+0 -10+0 236-293
=
 -146 -93 -243 -131 -90 -194 -50 -10 -57

Now, A^-1 = 1/319(A^2-A-293I)

:. A^-1 = 1/(319)
 -146 -93 -243 -131 -90 -194 -50 -10 -57

Solution provided by AtoZmath.com
Any wrong solution, solution improvement, feedback then Submit Here
Want to know about AtoZmath.com and me