Please could you help me with the below question. There are three parts, and all of my working is displayed! Thank you in advance, kind stranger.
For an integer n, real numbers a,b,c and an nxn matrix A for which: $ A^3+aA^2+bA+cI_n =0_n $
a) Assume that c does not equal 0, show A is invertible
Invertible => an nxn matrix B exists such that AB=In=BA. I found B to be: $ B=\frac{-1}{c}(A^2+aA+bI_n) $
b) Asume c=0 and b does not equal 0. Show A is invertible if and only if $ A^2+aA+bI_n=0 $
Assuming A is invertible: $$ A^{-1}(A^3+aA^2+bA)=A^{-1}0_N => A^2+aA+bI_n=0_n$$
Assuming $ A^2+aA+bI_n=0 $:
SOLVED - thank you Michael Hoppe and copper.hat
c) Write down two 3x3 matrices B which satisfy: $B^3-2B^2-6B+6I_3=0_n$ I can see that a=-2, b=-6 and c=6 and from part a), we know that B must be invertible. From here, I do to not know how to continue.
b) follows form the same considerations as a); $A(A+aI) = -b I$, so $A(-{1 \over b}(A+aI)) = I$.
For c), note that $f(x) = x^3-2x^2-6x+6$ has three distinct real roots ($f(-3)<0, f(2)>0, f(1)<0, f(4)>0$), call them $\lambda_1,\lambda_2, \lambda_3$.
Let $B_1 = \lambda_1 I = \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_1 & 0 \\ 0 & 0 & \lambda_1 \end{bmatrix}$, $B_2 = \lambda_2 I= \begin{bmatrix} \lambda_2 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & \lambda_2 \end{bmatrix}$.
Since each $\lambda_k$ satisfies $f(\lambda_k) = 0$, we see that $f(B_i) = 0$ too.