Question

The Cayley-Hamilton Theorem states that a matrix satisfies its characteristic equation. For example, the characteristic equation of the matrix shown below is as follows. 1-3 A = 12 - 61 + 11 = 0 and by the theorem you have A2 - 64 + 1112 = 0 2 5 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 0 5 -1 -1 3 1 0 0 1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of A. A2= A3 = STEP 3: Write a matrix version of the characteristic equation by replacing , with A. (Use I for the 3x3 identity matrix.) STEP 4: Substitute the powers of A into the matrix equation from step 3, and simplify. Is the matrix equation true? O Yes O No Need Help? Read It Talk to a Tutor

Answer 1

Here the given matrix A is A = 5 -1 3 1 L o The Characteristic equation of the matrix A is det (A-di) = 0 5 3rd 1 1-d! o or (1-2) 1-2 l=0 (1-d) {-1(3-1) +5} = 0 (1-d) (22 - 31+5) d²3d+5-d3+3 d²-5d=0 23 - 4 d & &d -5=0 The characteristic ecuation is 3-4 d² +82-5=0 Step 1 Step 2 Now, 5 -1 5 -1 3 1 A2 2 a 0 1 15 4 -3 4 5 1

A = A². 5-1 1-1 3 1 11 15 4 4 5 o -3 - 15 20 24 12 1 Step 3 The matrix version of the matrix A is A - 4A² + 8 A-513 - 0 step4 Now, A²-4A² + 8A-513 1-15 20 24 - 4 -3 12 1-5 15 4 olo 1+ 8 To 5-1 - 3 1 0 0 L .4-3 4 5 0 1 1-20 60 16 -12 16 20 To Go-8 1-8 24 8 ने -15 -4 0 20 24 12 0 1 1 - 15+20+40-5 1-4+12-8 2009-60 tho -3-16 +24-5 24-16-8 12-2018 1-448-5 o o O des