Enter a calculator syntax form eg px2 x3 qx r of det A x

Enter a calculator syntax form. (e.g. px^2 - x^3 + qx + r). of det (A - xI). the characters polynomial of the A = where A = [17/2 8 19/2 1 3 1 -13/2 -8 -15/2] The characteristic polynomial is The eigenvalues of A are Enter the eigenvalues as a comma-separated list

Solution

                     17/2            8             19/2

A       =            1              3                1

                    -13/2          -8             -15/2

The characterstic polynomial is =>

                         

( A - xI )    =    >

                    17/2            8             19/2                      1               0             0

                       1              3                1              - x  (   0              1              0     )          = 0

                    -13/2          -8             -15/2                     0              0              1

we get

                    17/2 - x            8              19/2

                       1                 3 - x              1                    =         0

                     -13/2               -8              -15/2 - x

.

(17/2 - x) [(3-x) (-15/2 - x) - 1 * (-8) ]    - 8 [1 * (-15/2 - x) + 13/2] + 19/2 [1 * (-8) +(13/2) * (3 - x) ] = 0

on solving this we get

         - x^3 + 4x^2 - x - 6   = 0                   // this is our characteristic polynomial

           (-1)(x + 1)(x - 2)(x - 3)           =    0

               (x + 1)(x - 2)(x - 3)           =    0

On solving these we get eigenvalues for the matrix A

               x = -1 , 2 , 3              // eigenvalues