1. **Problem Statement:** Given the matrix $$A = \begin{bmatrix} 3 & 1 \\ 2 & 1 \end{bmatrix},$$ we need to compute the following: (a) $$A^3$$ (b) $$A^{-3}$$ (c) $$A^2 - 2A + I$$ where $$I$$ is the identity matrix (d) $$p(A)$$ for $$p(x) = x - 2$$ (e) $$p(A)$$ for $$p(x) = 2x^2 - x + 1$$ (f) $$p(A)$$ for $$p(x) = x^3 - 2x + 4$$ 2. **Important Rules:** - Matrix multiplication is associative but not commutative. - Powers of a matrix are computed by repeated multiplication. - The identity matrix $$I$$ for 2x2 is $$\begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$. - For polynomial $$p(x)$$, $$p(A)$$ means substituting matrix $$A$$ into the polynomial, respecting matrix operations. - The inverse $$A^{-1}$$ satisfies $$AA^{-1} = I$$. 3. **Calculate $$A^2$$:** $$A^2 = A \times A = \begin{bmatrix}3 & 1 \\ 2 & 1\end{bmatrix} \times \begin{bmatrix}3 & 1 \\ 2 & 1\end{bmatrix} = \begin{bmatrix}3\cdot3 + 1\cdot2 & 3\cdot1 + 1\cdot1 \\ 2\cdot3 + 1\cdot2 & 2\cdot1 + 1\cdot1\end{bmatrix} = \begin{bmatrix}11 & 4 \\ 8 & 3\end{bmatrix}$$ 4. **Calculate $$A^3$$:** $$A^3 = A^2 \times A = \begin{bmatrix}11 & 4 \\ 8 & 3\end{bmatrix} \times \begin{bmatrix}3 & 1 \\ 2 & 1\end{bmatrix} = \begin{bmatrix}11\cdot3 + 4\cdot2 & 11\cdot1 + 4\cdot1 \\ 8\cdot3 + 3\cdot2 & 8\cdot1 + 3\cdot1\end{bmatrix} = \begin{bmatrix}41 & 15 \\ 30 & 11\end{bmatrix}$$ 5. **Calculate $$A^{-1}$$:** The inverse of a 2x2 matrix $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ is $$\frac{1}{ad - bc} \begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$$. For $$A$$, determinant $$= 3\cdot1 - 1\cdot2 = 3 - 2 = 1$$. So, $$A^{-1} = \begin{bmatrix}1 & -1 \\ -2 & 3\end{bmatrix}$$ 6. **Calculate $$A^{-3} = (A^{-1})^3$$:** First, compute $$A^{-2} = A^{-1} \times A^{-1} = \begin{bmatrix}1 & -1 \\ -2 & 3\end{bmatrix} \times \begin{bmatrix}1 & -1 \\ -2 & 3\end{bmatrix} = \begin{bmatrix}1\cdot1 + (-1)(-2) & 1\cdot(-1) + (-1)\cdot3 \\ -2\cdot1 + 3\cdot(-2) & -2\cdot(-1) + 3\cdot3\end{bmatrix} = \begin{bmatrix}3 & -4 \\ -8 & 11\end{bmatrix}$$ Then, $$A^{-3} = A^{-2} \times A^{-1} = \begin{bmatrix}3 & -4 \\ -8 & 11\end{bmatrix} \times \begin{bmatrix}1 & -1 \\ -2 & 3\end{bmatrix} = \begin{bmatrix}3\cdot1 + (-4)(-2) & 3\cdot(-1) + (-4)\cdot3 \\ -8\cdot1 + 11\cdot(-2) & -8\cdot(-1) + 11\cdot3\end{bmatrix} = \begin{bmatrix}11 & -15 \\ -30 & 41\end{bmatrix}$$ 7. **Calculate $$A^2 - 2A + I$$:** $$-2A = -2 \times \begin{bmatrix}3 & 1 \\ 2 & 1\end{bmatrix} = \begin{bmatrix}-6 & -2 \\ -4 & -2\end{bmatrix}$$ Sum: $$A^2 - 2A + I = \begin{bmatrix}11 & 4 \\ 8 & 3\end{bmatrix} + \begin{bmatrix}-6 & -2 \\ -4 & -2\end{bmatrix} + \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}11 - 6 + 1 & 4 - 2 + 0 \\ 8 - 4 + 0 & 3 - 2 + 1\end{bmatrix} = \begin{bmatrix}6 & 2 \\ 4 & 2\end{bmatrix}$$ 8. **Calculate $$p(A)$$ for $$p(x) = x - 2$$:** $$p(A) = A - 2I = \begin{bmatrix}3 & 1 \\ 2 & 1\end{bmatrix} - 2 \times \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}3 - 2 & 1 - 0 \\ 2 - 0 & 1 - 2\end{bmatrix} = \begin{bmatrix}1 & 1 \\ 2 & -1\end{bmatrix}$$ 9. **Calculate $$p(A)$$ for $$p(x) = 2x^2 - x + 1$$:** $$2A^2 = 2 \times \begin{bmatrix}11 & 4 \\ 8 & 3\end{bmatrix} = \begin{bmatrix}22 & 8 \\ 16 & 6\end{bmatrix}$$ $$-A = -1 \times \begin{bmatrix}3 & 1 \\ 2 & 1\end{bmatrix} = \begin{bmatrix}-3 & -1 \\ -2 & -1\end{bmatrix}$$ $$+I = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}$$ Sum: $$p(A) = 2A^2 - A + I = \begin{bmatrix}22 & 8 \\ 16 & 6\end{bmatrix} + \begin{bmatrix}-3 & -1 \\ -2 & -1\end{bmatrix} + \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix} = \begin{bmatrix}22 - 3 + 1 & 8 - 1 + 0 \\ 16 - 2 + 0 & 6 - 1 + 1\end{bmatrix} = \begin{bmatrix}20 & 7 \\ 14 & 6\end{bmatrix}$$ 10. **Calculate $$p(A)$$ for $$p(x) = x^3 - 2x + 4$$:** $$x^3 = A^3 = \begin{bmatrix}41 & 15 \\ 30 & 11\end{bmatrix}$$ $$-2x = -2A = \begin{bmatrix}-6 & -2 \\ -4 & -2\end{bmatrix}$$ $$+4I = \begin{bmatrix}4 & 0 \\ 0 & 4\end{bmatrix}$$ Sum: $$p(A) = A^3 - 2A + 4I = \begin{bmatrix}41 & 15 \\ 30 & 11\end{bmatrix} + \begin{bmatrix}-6 & -2 \\ -4 & -2\end{bmatrix} + \begin{bmatrix}4 & 0 \\ 0 & 4\end{bmatrix} = \begin{bmatrix}41 - 6 + 4 & 15 - 2 + 0 \\ 30 - 4 + 0 & 11 - 2 + 4\end{bmatrix} = \begin{bmatrix}39 & 13 \\ 26 & 13\end{bmatrix}$$ **Final answers:** (a) $$A^3 = \begin{bmatrix}41 & 15 \\ 30 & 11\end{bmatrix}$$ (b) $$A^{-3} = \begin{bmatrix}11 & -15 \\ -30 & 41\end{bmatrix}$$ (c) $$A^2 - 2A + I = \begin{bmatrix}6 & 2 \\ 4 & 2\end{bmatrix}$$ (d) $$p(A) = \begin{bmatrix}1 & 1 \\ 2 & -1\end{bmatrix}$$ (e) $$p(A) = \begin{bmatrix}20 & 7 \\ 14 & 6\end{bmatrix}$$ (f) $$p(A) = \begin{bmatrix}39 & 13 \\ 26 & 13\end{bmatrix}$$