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November 3rd, 2015, 08:46 AM  #1 
Newbie Joined: Nov 2015 From: Hungary Posts: 1 Thanks: 0  Det. of an n x n matrix, pls help to solve this problem
My task is to prove if it's true that: a.) det( A^2 + A + I(n) )>=0 b.) det( I(n) + A + B + A^2 +B^2) >= 0 if A and B are n x n matrices and A,B(i,j) takes the values from R (set of real numbers) and I(n) it's an n x n Identity matrix * >= means equal or higher Thanks 
December 27th, 2015, 02:34 PM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
`Hint: $\displaystyle A^2+ A+ I= A^2+ A+ \frac{1}{4}I+ \frac{3}{4}I= (A+ \frac{1}{2}I)^2+ \frac{3}{4}I$ $\displaystyle A^2+ B^2+ A+ B+ I= A^2+ A+ \frac{1}{4}I+ B^2+ B+ \frac{1}{4}I+ \frac{1}{2}I= (A+ \frac{1}{2}I)^2+ (B+ \frac{1}{2}I)^2+ \frac{1}{2}I$ 

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