Let's begin with **part (a)**, as you requested two steps for each subpart with explanations: --- ## (a) Find the characteristic polynomial and eigenvalues of matrix \( A \). **Step 1: Find the Characteristic Polynomial of \( A \).** The characteristic polynomial of a matrix \( A \) is found by computing \( \det(A - \lambda I) \): \[ A = \begin{bmatrix} 2 & 1 & 1 \\ -1 & 4 & 1 \\ 1 & -1 & 2 \\ \end{bmatrix} \] \[ A - \lambda I = \begin{bmatrix} 2-\lambda & 1 & 1 \\ -1 & 4-\lambda & 1 \\ 1 & -1 & 2-\lambda \\ \end{bmatrix} \] Now, compute the determinant: \[ \begin{align*} \det(A - \lambda I) =& (2-\lambda)[(4-\lambda)(2-\lambda) - (1)(-1)] \\ & - 1[-1(2-\lambda) - 1(1)] \\ & + 1[-1(4-\lambda) - (1)(1)] \\ \end{align*} \] Calculate each term: - First term: \((2-\lambda)[(4-\lambda)(2-\lambda) + 1]\) - Second term: \(-1[-(2-\lambda) - 1]\) - Third term: \(1[-(4-\lambda) - 1]\) Expand: \[ (2-\lambda)[(8 - 4\lambda - 2\lambda + \lambda^2) + 1] = (2-\lambda)[9 - 6\lambda + \lambda^2] \] Second term: \[ -1[-(2-\lambda) - 1] = (2-\lambda) + 1 = 3 - \lambda \] Third term: \[ 1[-(4-\lambda) - 1] = -4 + \lambda - 1 = \lambda - 5 \] Combine: \[ (2-\lambda)[9 - 6\lambda + \lambda^2] + (3 - \lambda) + (\lambda - 5) \] Simplify: \[ (2-\lambda)[9 - 6\lambda + \lambda^2] - 2 \] Now expand \( (2-\lambda)(9-6\lambda+\lambda^2) \): \[ = 2 \cdot (9 - 6\lambda + \lambda^2) - \lambda(9 - 6\lambda + \lambda^2) = 18 - 12\lambda + 2\lambda^2 - 9\lambda + 6\lambda^2 - \lambda^3 = 18 - 21\lambda + 8\lambda^2 - \lambda^3 \] So, the characteristic polynomial is: \[ \boxed{-\lambda^3 + 8\lambda^2 - 21\lambda + 16} \] --- **Step 2: Find the Eigenvalues of \( A \) (Roots of the Characteristic Polynomial).** Set the polynomial to zero: \[ -\lambda^3 + 8\lambda^2 - 21\lambda + 16 = 0 \] Or \[ \lambda^3 - 8\lambda^2 + 21\lambda - 16 = 0 \] Let's try \(\lambda = 1\): \[ 1^3 - 8\cdot1^2 + 21\cdot1 - 16 = 1 - 8 + 21 - 16 = -7 + 21 - 16 = 14 - 16 = -2 \neq 0 \] Try \(\lambda = 2\): \[ 8 - 32 + 42 - 16 = (8 + 42) - (32 + 16) = 50 - 48 = 2 \neq 0 \] Try \(\lambda = 4\): \[ 64 - 128 + 84 - 16 = (64 + 84) - (128 + 16) = 148 - 144 = 4 \neq 0 \] Try \(\lambda = 8\): \[ 512 - 512 + 168 - 16 = 0 + 168 - 16 = 152 \neq 0 \] Try \(\lambda = 1\) again (but let's check \( \lambda = 1 \)): \[ 1^3 - 8 \cdot 1^2 + 21 \cdot 1 - 16 = 1 - 8 + 21 - 16 = -7 + 21 - 16 = 14 - 16 = -2 \] Try \(\lambda = 16\): \[ 16^3 - 8 \cdot 16^2 + 21 \cdot 16 - 16 = 4096 - 2048 + 336 - 16 = 2048 + 336 - 16 = 2384 - 16 = 2368 \] So, none of these small integer values work. Try \(\lambda = 1, 2, 4, 8\) (by Rational Root Theorem), but none of these work. Let's check for \(\lambda = 1\) again: \[ 1 - 8 + 21 - 16 = -7 + 21 - 16 = 14 - 16 = -2 \] No. Try \(\lambda = 16\): \[ 4096 - 2048 + 336 - 16 = 2048 + 336 - 16 = 2384 - 16 = 2368 \] No. Try \(\lambda = 1, 2, 4, 8\), as divisors of 16. But none work. Let's perform synthetic division or factorization. Alternatively, we can use the cubic formula or numerical methods, but for now, let's factor: Try \(\lambda = 1, 2, 4, 8\) as possible rational roots, but none of these fit. Let's look for quadratic factors. Alternatively, use the fact that the trace (sum of eigenvalues) is 8, and determinant (product) is 16. The polynomial is: \[ \lambda^3 - 8\lambda^2 + 21\lambda - 16 \] Let's factor: Try \((\lambda - 1)(\lambda^2 - 7\lambda + 16)\): \[ (\lambda - 1)(\lambda^2 - 7\lambda + 16) = \lambda^3 - 7\lambda^2 + 16\lambda - \lambda^2 + 7\lambda - 16 = \lambda^3 - 8\lambda^2 + 23\lambda - 16 \] But that's \(+23\lambda\), not \(+21\lambda\). Try \((\lambda - 2)(\lambda^2 - 6\lambda + 8)\): \[ (\lambda - 2)(\lambda^2 - 6\lambda + 8) = \lambda^3 - 6\lambda^2 + 8\lambda -2\lambda^2 + 12\lambda - 16 = \lambda^3 - 8\lambda^2 + 20\lambda - 16 \] That's \(20\lambda\), not \(21\lambda\). Try \((\lambda - 4)(\lambda^2 - 4\lambda + 4)\): \[ (\lambda - 4)(\lambda^2 - 4\lambda + 4) = \lambda^3 - 4\lambda^2 + 4\lambda - 4\lambda^2 + 16\lambda - 16 = \lambda^3 - 8\lambda^2 + 20\lambda - 16 \] Again \(20\lambda\). Try \((\lambda - a)(\lambda^2 - b\lambda + c)\) and match coefficients: Expand: \[ \lambda^3 - b\lambda^2 + c\lambda - a\lambda^2 + ab\lambda - ac = \lambda^3 - (a+b)\lambda^2 + (ab+c)\lambda - ac \] Set equal to: \[ \lambda^3 - 8\lambda^2 + 21\lambda - 16 \] So, \[ a + b = 8 \\ ab + c = 21 \\ ac = 16 \\ \] Try \(a = 2\): \[ b = 6, c = 21 - ab = 21 - 12 = 9, ac = 2*9 = 18 \neq 16 \] Try \(a = 4\): \[ b = 4, c = 21 - 16 = 5, ac = 4*5 = 20 \neq 16 \] Try \(a = 8\): \[ b = 0, c = 21 - 0 = 21, ac = 8*21 = 168 \neq 16 \] Try \(a = 1\): \[ b = 7, c = 21 - 7 = 14, ac = 1*14 = 14 \neq 16 \] Try \(a = 16\): \[ b = -8, c = 21 - (-128) = 149, ac = 16*149 = 2384 \neq 16 \] Try \(a = 8, b = 0\), c = 21 - 0 = 21, ac = 8*21 = 168 Try \(a = 16, b = -8\), c = 21 - (-128) = 149, ac = 16*149 = 2384 Try \(a = 16, b = -8, c = 21 - (-128) = 149, ac = 16*149 = 2384\) No good rational roots. Let's use the cubic formula or estimate roots numerically. **Summary:** - **Characteristic polynomial:** \(\boxed{\lambda^3 - 8\lambda^2 + 21\lambda - 16}\) - **Eigenvalues:** Roots of the above cubic (approximate values or factorization needed for exact values). --- **Final Answer for (a):** 1. **Characteristic Polynomial:** \(\boxed{\lambda^3 - 8\lambda^2 + 21\lambda - 16}\) 2. **Eigenvalues:** Roots of the cubic above (approximate values or further factorization required). --- Let me know if you want to continue to part (b) or want a deeper factorization of the roots!