單選4
設矩陣\(A=\left[\matrix{-5&-4\cr 9&7}\right]\)，則\(A^{51}-A^{50}+A^3-3A^2-2A+4I_2\)為下列何者？(\(I_2=\left[\matrix{1&0 \cr 0&1}\right]\))
(A)\(\left[\matrix{24&16 \cr -36&-24}\right]\) (B)\(\left[\matrix{-24&-16\cr 36&24}\right]\) (C)\(O_2\) (D)\(4I_2\)
[解答]
原式=(x-1)[(x-1)+1]^50+(x-1)^3-5x+5
=(x-1)[(x-1)Q(x)+1]+(x-1)^3-5x+5
=(x-1)[(x-1)Q(x)+1]+(x-1)^3-5x+5
=(x-1)^2*Q(x)-4x+4
所求=-4A+4I= (A)選項
