Characteristic Roots Of The Matrix

Question

I am supposed to find the characteristic roots of the matrix

X=
(-12 5
5 2)
... how should I do that

Answer 1

I assume you mean find the eigenvalues of the matrix.  The eigenvalues are the roots of the equation

 

det(A - λI) = 0

 

where A is the given square matrix, I is the identity matrix, and "det" is the determinant.  In this problem,

 

A = |-12  5|

      |   5  2|

 

A - λI =  |-12 - λ           5|
              | 5            2 - λ|

 

det(A - λI) = (-12 - λ)(2 - λ) - (5)(5) = 0

 

That is your characteristic equation.  Solve the resulting quadratic equation for λ and the values of λ will be your characteristic roots or eigenvalues.