1. **State the problem:** We need to show that $$A^3 - A^2 + A - 2B = 0$$ where $$A = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}$$ and $$B = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$$. 2. **Calculate $$A^2$$:** $$A^2 = A \times A = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} = \begin{bmatrix} (1)(1)+(0)(1)+(1)(0) & (1)(0)+(0)(1)+(1)(1) & (1)(1)+(0)(0)+(1)(1) \\ (1)(1)+(1)(1)+(0)(0) & (1)(0)+(1)(1)+(0)(1) & (1)(1)+(1)(0)+(0)(1) \\ (0)(1)+(1)(1)+(1)(0) & (0)(0)+(1)(1)+(1)(1) & (0)(1)+(1)(0)+(1)(1) \end{bmatrix}$$ $$= \begin{bmatrix} 1 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix}$$. 3. **Calculate $$A^3$$:** $$A^3 = A \times A^2 = \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} \times \begin{bmatrix} 1 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix} = \begin{bmatrix} (1)(1)+(0)(2)+(1)(1) & (1)(1)+(0)(1)+(1)(2) & (1)(2)+(0)(1)+(1)(1) \\ (1)(1)+(1)(2)+(0)(1) & (1)(1)+(1)(1)+(0)(2) & (1)(2)+(1)(1)+(0)(1) \\ (0)(1)+(1)(2)+(1)(1) & (0)(1)+(1)(1)+(1)(2) & (0)(2)+(1)(1)+(1)(1) \end{bmatrix}$$ $$= \begin{bmatrix} 2 & 3 & 3 \\ 3 & 2 & 3 \\ 3 & 3 & 2 \end{bmatrix}$$. 4. **Calculate the expression:** $$A^3 - A^2 + A - 2B = \begin{bmatrix} 2 & 3 & 3 \\ 3 & 2 & 3 \\ 3 & 3 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} - 2 \times \begin{bmatrix} 0 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}$$ $$= \begin{bmatrix} 2 & 3 & 3 \\ 3 & 2 & 3 \\ 3 & 3 & 2 \end{bmatrix} - \begin{bmatrix} 1 & 1 & 2 \\ 2 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 0 & 1 \\ 1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix} - \begin{bmatrix} 0 & 2 & 2 \\ 2 & 2 & 2 \\ 2 & 2 & 2 \end{bmatrix}$$ 5. **Perform element-wise subtraction and addition:** Calculate $$\begin{bmatrix} 2-1+1-0 & 3-1+0-2 & 3-2+1-2 \\ 3-2+1-2 & 2-1+1-2 & 3-1+0-2 \\ 3-1+0-2 & 3-2+1-2 & 2-1+1-2 \end{bmatrix}$$ $$= \begin{bmatrix} 2 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$. 6. Note that this is not the zero matrix; therefore the original claim $$A^3 - A^2 + A - 2B = 0$$ is **not true** with the provided matrices. **Final answer:** The expression does not equal the zero matrix; hence, the statement is false. Note: The calculation showed that $$(A^3 - A^2 + A - 2B)_{1,1} = 2 \neq 0$$ verifying the expression fails to hold. 7. The initial equations $$2p^2 + q^2 = 3$$ and $$p + q = 2$$ and the addition $(5 + 5 = 10)$ are unrelated to this matrix problem.