Let's solve the given problem step by step. Given matrix: \[ A = \begin{bmatrix} 1 & 0 & -1 \\ 1 & -3 & 0 \\ 4 & -13 & 1 \end{bmatrix} \] --- ## **(a) Find all eigenvalues of A** The eigenvalues \(\lambda\) are found by solving: \[ \det(A - \lambda I) = 0 \] First, compute \(A - \lambda I\): \[ A - \lambda I = \begin{bmatrix} 1-\lambda & 0 & -1 \\ 1 & -3-\lambda & 0 \\ 4 & -13 & 1-\lambda \end{bmatrix} \] Now, compute the determinant: \[ \det(A - \lambda I) = \left| \begin{array}{ccc} 1-\lambda & 0 & -1 \\ 1 & -3-\lambda & 0 \\ 4 & -13 & 1-\lambda \end{array} \right| \] We'll use the first row for expansion: \[ = (1-\lambda) \left| \begin{array}{cc} -3-\lambda & 0 \\ -13 & 1-\lambda \\ \end{array} \right| - 0 \cdot \left| \begin{array}{cc} 1 & 0 \\ 4 & 1-\lambda \\ \end{array} \right| + (-1) \left| \begin{array}{cc} 1 & -3-\lambda \\ 4 & -13 \\ \end{array} \right| \] Let's compute each piece: 1. \( (1-\lambda) \) minor: \[ \left| \begin{array}{cc} -3-\lambda & 0 \\ -13 & 1-\lambda \\ \end{array} \right| = (-3-\lambda)(1-\lambda) - (0)(-13) = (-3-\lambda)(1-\lambda) \] 2. The \( -1 \) minor: \[ \left| \begin{array}{cc} 1 & -3-\lambda \\ 4 & -13 \\ \end{array} \right| = 1 \cdot (-13) - 4(-3-\lambda) = -13 + 12 + 4\lambda = (-13+12) + 4\lambda = -1 + 4\lambda \] So, the determinant is: \[ \det(A-\lambda I) = (1-\lambda)[(-3-\lambda)(1-\lambda)] - [ -1 + 4\lambda ] \] But we must not forget the 0 coefficient in the expansion! Let's write the full expansion: \[ \det(A-\lambda I) = (1-\lambda)[(-3-\lambda)(1-\lambda) - (0)(-13)] - 0 \cdot [1 \cdot (1-\lambda) - 0 \cdot 4] + (-1)[1 \cdot (-13) - 4 \cdot (-3-\lambda)] \] \[ = (1-\lambda)[(-3-\lambda)(1-\lambda)] + (1)[-13 + 4(3+\lambda)] \] \[ = (1-\lambda)[(-3-\lambda)(1-\lambda)] + [-13 + 12 + 4\lambda] \] \[ = (1-\lambda)[(-3-\lambda)(1-\lambda)] + (-1 + 4\lambda) \] Let's expand \((1-\lambda)(-3-\lambda)(1-\lambda)\): First, expand \((1-\lambda)(-3-\lambda)\): \[ (1-\lambda)(-3-\lambda) = 1 \cdot (-3-\lambda) - \lambda(-3-\lambda) = -3 - \lambda + 3\lambda + \lambda^2 = (-3 + 2\lambda + \lambda^2) \] Therefore, \[ \det(A-\lambda I) = (1-\lambda)(-3 + 2\lambda + \lambda^2) + (-1 + 4\lambda) \] Now expand \((1-\lambda)(-3 + 2\lambda + \lambda^2)\): \[ (1-\lambda)(-3 + 2\lambda + \lambda^2) = 1 \cdot (-3 + 2\lambda + \lambda^2) - \lambda(-3 + 2\lambda + \lambda^2) = (-3 + 2\lambda + \lambda^2) - [-3\lambda + 2\lambda^2 + \lambda^3] \] \[ = -3 + 2\lambda + \lambda^2 + 3\lambda - 2\lambda^2 - \lambda^3 \] \[ = -3 + (2\lambda + 3\lambda) + (\lambda^2 - 2\lambda^2) - \lambda^3 \] \[ = -3 + 5\lambda - \lambda^2 - \lambda^3 \] Add \(-1 + 4\lambda\): \[ \det(A-\lambda I) = [-3 + 5\lambda - \lambda^2 - \lambda^3] + (-1 + 4\lambda) \] \[ = (-3 - 1) + (5\lambda + 4\lambda) - \lambda^2 - \lambda^3 \] \[ = -4 + 9\lambda - \lambda^2 - \lambda^3 \] So, \[ \boxed{ \det(A-\lambda I) = -\lambda^3 - \lambda^2 + 9\lambda - 4 } \] Or, factor out the minus sign: \[ \lambda^3 + \lambda^2 - 9\lambda + 4 = 0 \] **Now, solve for the roots (eigenvalues) of:** \[ \lambda^3 + \lambda^2 - 9\lambda + 4 = 0 \] Let's use the Rational Root Theorem. Possible rational roots are \(\pm1, \pm2, \pm4\). Test \(\lambda=1\): \[ 1^3 + 1^2 - 9 \cdot 1 + 4 = 1 + 1 - 9 + 4 = 2 - 9 + 4 = -7 + 4 = -3 \neq 0 \] Test \(\lambda=-1\): \[ (-1)^3 + (-1)^2 - 9(-1) + 4 = -1 + 1 + 9 + 4 = 13 \] Test \(\lambda=2\): \[ 8 + 4 - 18 + 4 = 12 - 18 + 4 = -6 + 4 = -2 \] Test \(\lambda=-2\): \[ -8 + 4 + 18 + 4 = -8 + 26 = 18 + 4 = 22 \] Test \(\lambda=4\): \[ 64 + 16 - 36 + 4 = 80 - 36 + 4 = 44 + 4 = 48 \] Wait. Let's check carefully: the cubic is \(\lambda^3 + \lambda^2 - 9\lambda + 4\). So for \(\lambda=4\): \[ 4^3 + 4^2 - 9 \cdot 4 + 4 = 64 + 16 - 36 + 4 = 80 - 36 + 4 = 44 + 4 = 48 \] Test \(\lambda=-4\): \[ (-4)^3 + (-4)^2 - 9(-4) + 4 = -64 + 16 + 36 + 4 = -64 + 56 = -8 + 4 = -4 \] Try \(\lambda=0.5\): \[ (0.5)^3 + (0.5)^2 - 9 \cdot (0.5) + 4 = 0.125 + 0.25 - 4.5 + 4 = 0.375 - 4.5 + 4 = -4.125 + 4 = -0.125 \] Try \(\lambda=4\), as above, was 48. Try \(\lambda=0.25\): \[ (0.25)^3 + (0.25)^2 - 9 \cdot 0.25 + 4 = 0.015625 + 0.0625 - 2.25 + 4 = 0.078125 - 2.25 + 4 = -2.171875 + 4 = 1.828125 \] Try \(\lambda=0\): \[ 0 + 0 - 0 + 4 = 4 \] Try \(\lambda=1\): As above, -3. Try \(\lambda=