SOLUTION: Let A represent a matrix. Show that if A^3 - 7A^2 + 5In = 0 then A^4 = 49A^2 - 5A - 35In

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Question 161620This question is from textbook Primer of Linear Algebra
: Let A represent a matrix.
Show that if A^3 - 7A^2 + 5In = 0 then A^4 = 49A^2 - 5A - 35In This question is from textbook Primer of Linear Algebra

Answer by kev82(151) About Me  (Show Source):
You can put this solution on YOUR website!
I am going to use I instead of In for the the identity. So we are given that A^3 - 7A^2 + 5I = 0 therefore A^3 = 7A^2 - 5I.
We are interested in A^4 which is A*A^3, but we know what A^3 is, so substitute A^4 = A(7A^2-5I) = 7A^3-5A. WE now have another A^3 we can substitute for, so do that. A^4 = 7(7A^2-5I)-5A = 49A^2-5A-35I.