1 vote

if $\lambda$^{3} - 6$\lambda$^{2} -$\lambda$ +22=0 is a characteristic of 3 X 3 diagonal matrix , then trace of matrix A is

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Best answer

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 : λ^{3} - 6λ^{2} - λ + 22 = 0

We know sum of roots here = - (Coefficient of λ^{2 }) / (Coefficient of λ^{3 })

= 6

**Hence trace of matrix = 6**