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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:: 22 - 61 + 11 = 0 and by the theorem you have 42 - 64 + 1112 = 0 Demonstrate the Cayley-Hamilton Theorem for the matrix A given below. 03 1 A = -1 5 1 0 0 -1 STEP 1: Find and expand the characteristic equation. STEP 2: Compute the required powers of A. X X X 42 = x x 0 0 1 x x x A3 = X x 0 0 -1 STEP 3: Write a matrix version of the characteristic equation by replacing 2 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
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