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December 18th, 2017, 09:50 AM  #1 
Senior Member Joined: Jan 2017 From: Toronto Posts: 200 Thanks: 2  Finding matrix A inverse...
If A is an n x n matrix and it satisfies the equation: $\displaystyle A^3  4A^2 + 3A  5I_n = 0 $ then A is nonsingular and its inverse is ?????? 
December 18th, 2017, 10:52 AM  #2 
Senior Member Joined: Sep 2016 From: USA Posts: 520 Thanks: 293 Math Focus: Dynamical systems, analytic function theory, numerics 
Its inverse is $\frac{1}{5}\left( A^2 4A + 3 \right)$. Do you see why?

December 18th, 2017, 10:57 AM  #3 
Senior Member Joined: Sep 2015 From: USA Posts: 2,202 Thanks: 1156  
December 18th, 2017, 10:57 AM  #4 
Senior Member Joined: Jan 2017 From: Toronto Posts: 200 Thanks: 2  
December 18th, 2017, 11:21 AM  #5  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894  Quote:
Equivalently, $\displaystyle A^3 4A^2+ 3A 5I= 0$ so $\displaystyle A^3 4A^2+ 3A= 5I$ $\displaystyle A(A^2 4A+ 2)= (A^2 4A+ 3)A= 5I$ $\displaystyle A((1/5)(A^2 4A+ 3))= ((1/5)(A^2 4A+ 3)A= I$. That is sufficient to show that $\displaystyle (1/5)(A^2 4A+ 3)$ is the inverse of A. Last edited by skipjack; December 18th, 2017 at 05:16 PM.  

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