trace of matrix

Question

If $\lambda^3 - 6\lambda^2 -\lambda +22=0$ is a characteristic equation of $3\ X\ 3$ diagonal matrix, then trace of matrix is

Answer 1

The following points to be noted :

a) In a diagonal matrix D , determinant(D)  =  a.b.c.... where each of a,b, c and so on are diagonal elements.

b) The diagonal elements of  the diagonal matrix are eigen values as well..

c) Trace is the sum of the diagonal elements of a matrix..

d) The sum of eigen values is given by trace of the matrix.

e) The roots of characteristic equation are nothing but eigen values of the corresponding matrix 

So , trace of a matrix is nothing but the sum of eigen values..

And hence given a characteristic equation , sum of eigen values is the same as sum of roots of the corresponding characteristic equation..

Given characteristic equation :    λ³ - 6λ² - λ + 22 = 0

We know sum of roots here   =  - (Coefficient of λ² ) / (Coefficient of  λ³ )

                                           =  6

Hence trace of matrix           =  6

Answer 2

( t1 – $\lambda$ ) ( t2 – $\lambda$ ) ( t3 – $\lambda$ ) =0 is the characteristic equation. In this the coeffecient of $-\lambda ^{2} $ gives the trace.