Homework SourseThe matrix A 1 12 k 13 0 0 1 0 has three distinct real eige
The matrix A = [1 12 k 1-3 0 0 1 0] has three distinct real eigenvalues if and only if ![The matrix A = [1 12 k 1-3 0 0 1 0] has three distinct real eigenvalues if and only if SolutionThe characteristic equation of A is det(A-I3) = 0 or, 3 +22-15 –](/WebImages/38/the-matrix-a-1-12-k-13-0-0-1-0-has-three-distinct-real-eige-1116836-1761593412-0.webp "The matrix A = [1 12 k 1-3 0 0 1 0] has three distinct real eigenvalues if and only if SolutionThe characteristic equation of A is det(A-I3) = 0 or, 3 +22-15 –")
Solution
The characteristic equation of A is det(A-I3) = 0 or, 3 +22-15 –k = 0. Now, we know that the cubic equation Ax3 +Bx2 +Cx +D = 0 has 3 real roots if -27A2 D2 +18ABCD -4AC3-4B3D +B2 C2 > 0. Here, A = 1, B = 2, C = -15 and D = -k . Hence the equation 3 +22-15 –k = 0 will have 3 real roots if -27{12(-k)2} +18(1)(2)(-15)(-k) -4(1)(-15)3 -4(2)3(-k) +(2)2( -15)2 > 0 or, -27k2+ 540k +13500+32k + 900> 0 or, -27k2+ 572k+14400 > 0 or, -27k2+ 572k > -14400 or, 27k2 -572k < 14400 . Thus, A will have 3 distinct real roots if and only if -400/27 < k < 36. ( -400/27 and 36 are the roots of the quadratic 27k2 -572k -14400 = 0)
