### Related Formula textInductive formulation for exponent powers of a pattern matrix$\text{Inductive formulation for exponent powers of a pattern matrix}$ ### Core Logic Evaluate lower power forms of A$A$ to establish inductive sequence patterns: A = beginbmatrix 2 & -1 \\ 1 & 0 endbmatrix, quad A^2 = beginbmatrix 3 & -2 \\ 2 & -1 endbmatrix, quad A^3 = beginbmatrix 4 & -3 \\ 3 & -2 endbmatrix$A = \begin{\bmatrix} 2 & -1 \\ 1 & 0 \end{\bmatrix}, \quad A^2 = \begin{\bmatrix} 3 & -2 \\ 2 & -1 \end{\bmatrix}, \quad A^3 = \begin{\bmatrix} 4 & -3 \\ 3 & -2 \end{\bmatrix}$ This cleanly establishes general power state rule configuration expression: A^m = beginbmatrix m+1 & -m \\ m & -m+1 endbmatrix$A^m = \begin{\bmatrix} m+1 & -m \\ m & -m+1 \end{\bmatrix}$ ### Step 1: Setup Matrix Power Equation Using the pattern, find expressions for terms: A^6 = beginbmatrix 7 & -6 \\ 6 & -5 endbmatrix, quad A^-6 = (A^6)^-1 = beginbmatrix -5 & 6 \\ -6 & 7 endbmatrix$A^6 = \begin{\bmatrix} 7 & -6 \\ 6 & -5 \end{\bmatrix}, \quad A^{-6} = (A^6)^{-1} = \begin{\bmatrix} -5 & 6 \\ -6 & 7 \end{\bmatrix}$ Substitute into the targeted equation block: A^m^2 + A^m = 3beginbmatrix 1 & 0 \\ 0 & 1 endbmatrix - beginbmatrix -5 & 6 \\ -6 & 7 endbmatrix = beginbmatrix 8 & -6 \\ 6 & -4 endbmatrix$A^{m^2} + A^m = 3\begin{\bmatrix} 1 & 0 \\ 0 & 1 \end{\bmatrix} - \begin{\bmatrix} -5 & 6 \\ -6 & 7 \end{\bmatrix} = \begin{\bmatrix} 8 & -6 \\ 6 & -4 \end{\bmatrix}$ ### Step 2: Equate Corresponding Elements From the bottom-\left entry [2,1]$[2,1]$: m^2 + m = 6 implies m^2 + m - 6 = 0$m^2 + m = 6 \implies m^2 + m - 6 = 0$ (m+3)(m-2) = 0 implies m = -3, 2$(m+3)(m-2) = 0 \implies m = -3, 2$ Both integer states satisfy all matrix components seamlessly. Therefore, the number of elements n(S) = 2$n(S) = 2$. ### Pattern Recognition Always calculate first 2–3 matrix powers to spot linear arithmetic trends across specific cell values (m+1, -m$m+1, -m$), avoiding tedious full Cayley-Hamilton character expansions. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 12 Mathematics: Matrices