Jun 11, 2013 Thread starter #1 D delgeezee New member May 24, 2013 9 Let A be a square matrix, a) show that \(\displaystyle (I-A)^{-1}= I + A + A^2 + A^3 if A^4 = 0\) b) show that \(\displaystyle (I-A)^{-1}= I + A + A^2+.....+A^n \) if \(\displaystyle A^{n+1}= 0\)
Let A be a square matrix, a) show that \(\displaystyle (I-A)^{-1}= I + A + A^2 + A^3 if A^4 = 0\) b) show that \(\displaystyle (I-A)^{-1}= I + A + A^2+.....+A^n \) if \(\displaystyle A^{n+1}= 0\)
Jun 11, 2013 #2 A Amer Active member Mar 1, 2012 275 Re: polynomial problem proof? B is the inverse of A iff [tex] AB = BA = I [/tex] so try [tex] (I -A )( I + A + A^2 + A^3) = I-A + A - A^2 + A^2 - A^3 + A^3 - A^4 = I - A^4 = I [/tex]
Re: polynomial problem proof? B is the inverse of A iff [tex] AB = BA = I [/tex] so try [tex] (I -A )( I + A + A^2 + A^3) = I-A + A - A^2 + A^2 - A^3 + A^3 - A^4 = I - A^4 = I [/tex]
