Use the Cayley-Hamilton theorem to write the matrix inverse A^-1 in terms of I,
ID: 3013046 • Letter: U
Question
Use the Cayley-Hamilton theorem to write the matrix inverse A^-1 in terms of I, A, A^2 for the following matrix A = [2 0 0 0 1 0 0 1 1]Explanation / Answer
Dear Student Thank you for using Chegg !! Given matrix A A = 2 0 0 0 1 0 0 1 1 Now characteristic equation is given by A-kI = 0 => 2-k 0 0 0 1-k 0 0 1 1-k => (k-1)^2(k-2)=0 k^3-4k^2 + 5k - 2 = 0 (Characteriestic Equation) Now as per Caley hamilton theorm, k can be substituted with given matrix A i.e. A^3 - 4A^2 + 5A -2I = 0 => A^2-4A+5I = 2A^(-1) Calculating A^2 2 0 0 2 0 0 4 0 0 A^2 = 0 1 0 X 0 1 0 = 0 1 0 0 1 1 0 1 1 0 2 1 -4A = -8 0 0 0 -4 0 0 -4 -4 5I = 5 0 0 0 5 0 0 0 5 A^2-4A+5I = 1 0 0 0 2 0 0 -2 2 A^-1 = (1/2) (A^2-4A+5I) A^-1 = 0.5 0 0 0 1 0 0 -1 1 Hence Proved
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