The limit state of the Markov processed model by the matrix [ 0.2 0.5 0.8 0.5 ] . The limit state of the Markov processed model by the matrix [ 0.2 0.5 0.8 0.5 ] is [ 5 8 ] _ . Definition used: Let A be a n × n matrix. The eigen values of the matrix A is obtained by solving the characteristic equation D ( λ ) = 0 that is det ( A − λ I ) = 0 , where λ is the eigen value. Calculation: The given matrix is [ 0.2 0.5 0.8 0.5 ] . Let A = [ 0.2 0.5 0.8 0.5 ] . Substitute A = [ 0.2 0.5 0.8 0.5 ] in det ( A − λ I ) = 0 and obtain the characteristic equation as shown below: | [ 0.2 0.5 0.8 0.5 ] − λ [ 1 0 0 1 ] | = 0 | 0.2 − λ 0.5 0.8 0.5 − λ | = 0 ( 0.2 − λ ) ( 0.5 − λ ) − 0.4 = 0 Solve the above equation and obtain the eigen values as follows: ( 0.2 − λ ) ( 0.5 − λ ) − 0.4 = 0 λ = 1 , − 0.3 Thus, the eigen values are λ = 1 , − 0.3 . Substitute λ = 1 in equation (1) as, [ A − λ I ] λ = 1 = [ 0.2 − ( 1 ) 0.5 0.8 0.5 − ( 1 ) ] = [ − 0.8 0.5 0.8 − 0.5 ] Let x = [ x 1 x 2 ] be the eigen vector. Solve the equation [ A − λ I ] x = 0 as follows: [ A − λ I ] x = 0 [ − 0.8 0.5 0.8 − 0.5 ] [ x 1 x 2 ] = 0 { − 0.8 x 1 + 0.5 x 2 = 0 0.8 x 1 − 0.5 x 2 = 0 { − 8 x 1 + 5 x 2 = 0 8 x 1 − 5 x 2 = 0 From the above system, the equation 8 x 1 = 5 x 2 can be obtained. Thus, the vector x can be written as follows: x = [ x 1 x 2 ] = [ 5 8 ] Therefore, the limit state is [ 5 8 ] _

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