Let's carefully solve this step by step manually. We are given a linear operator $T: \mathbb{R}^3 \to \mathbb{R}^3$ defined by $$ T(x, y, z) = (x+y, \; y, \; 2x - 2y + 2z) $$ and we are asked to check if $T$ satisfies the polynomial $$ p(x) = (x-1)^2 (x-2) $$ and then find the minimal polynomial of $T$. --- ### Step 1: Represent $T$ as a matrix Using the standard basis $e_1 = (1,0,0), e_2 = (0,1,0), e_3 = (0,0,1)$: * $T(e_1) = T(1,0,0) = (1+0, 0, 2 - 0 + 0) = (1,0,2)$ * $T(e_2) = T(0,1,0) = (0+1, 1, 0 - 2 + 0) = (1,1,-2)$ * $T(e_3) = T(0,0,1) = (0+0,0,0 - 0 + 2) = (0,0,2)$ So the matrix of $T$ in standard basis is: $$ [T] = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 2 & -2 & 2 \end{pmatrix} $$ --- ### Step 2: Check if $T$ satisfies $p(x) = (x-1)^2(x-2)$ We need to compute $(T-I)^2(T-2I)$ and check if it equals the zero matrix. #### Step 2a: Compute $T-I$ $$ T - I = \begin{pmatrix} 1-1 & 1 & 0 \\ 0 & 1-1 & 0 \\ 2 & -2 & 2-1 \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 2 & -2 & 1 \end{pmatrix} $$ #### Step 2b: Compute $(T-I)^2$ $$ (T-I)^2