The eigen values and eigen vectors of the system of equations. The eigen values of the system of equations are λ = 9 _ . The eigen vectors of the system of equations is [ 2 − 2 1 ] T , [ − 2 2 − 1 ] T , [ − 1 1 0.5 ] T _ . 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 [ 13 5 2 2 7 − 8 5 4 7 ] . Let A = [ 13 5 2 2 7 − 8 5 4 7 ] . Substitute A = [ 13 5 2 2 7 − 8 5 4 7 ] in det ( A − λ I ) = 0 and obtain the characteristic equation as shown below: | [ 13 5 2 2 7 − 8 5 4 7 ] − λ [ 1 0 0 0 1 0 0 0 1 ] | = 0 | 13 − λ 5 2 2 7 − λ − 8 5 4 7 − λ | = 0 ( ( 13 − λ ) [ ( 7 − λ ) ( 7 − λ ) ] − 5 [ 2 ( 7 − λ ) + 40 ] + 2 [ 8 − 5 ( 7 − λ ) ] = 0 ) Solve the above equation and obtain the eigen values as follows: ( ( 13 − λ ) [ ( 7 − λ ) ( 7 − λ ) ] − 5 [ 2 ( 7 − λ ) + 40 ] + 2 [ 8 − 5 ( 7 − λ ) ] = 0 ) ( λ − 9 ) ( λ − 9 ) ( λ − 9 ) = 0 λ = 9 , λ = 9 , λ = 9 Thus, the eigen values are λ = 9 _ . Substitute λ = 9 in [ A − λ I ] as, [ A − λ I ] λ = 9 = [ 13 − 9 5 2 2 7 − 9 − 8 5 4 7 − 9 ] = [ 4 5 2 2 − 2 6 5 4 − 2 ] Let x = [ x 1 x 2 x 3 ] be the eigen vector. Solve the equation [ A − λ I ] x = 0 as follows: [ A − λ I ] x = 0 ⇒ [ 4 5 2 2 − 2 6 5 4 − 2 ] [ x 1 x 2 x 3 ] = 0 ⇒ { 4 x 1 + 5 x 2 + 2 x 3 = 0 2 x 1 − 2 x 2 + 6 x 3 = 0 5 x 1 + 4 x 2 − 2 x 3 = 0 Thus, the three eigen vector x can be written as follows: x = [ x 1 x 2 x 3 ] = { [ 2 − 2 1 ] , [ − 2 − 2 − 1 ] , [ − 1 − 2 − 0.5 ] } Therefore, the eigen vector for λ = 1 is [ 2 − 2 1 ] T , [ − 2 2 − 1 ] T , [ − 1 1 0.5 ] T _ .

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