matrix maths

Question

1  -3   3

0  -5  6

0  -3   4    

a 3*3 matrix is given  if  x,y, z are the eigan value then find  xy+yz+ax?

 my approch… if i do row transformation in c2->c2+c3

and then c2->4c2-c3,  so my matrix become upper tringular matrix then the diagonal element are the EV so i am getting -6

but using genral method via substract  lemda from diagonal element and then determinant of matrix  getting answer -3

which one is correct and why not other one

Comments

@sachin_mittal sir?

Answer 1

Elementary row or column operation will change the eigenvalues of a matrix.

Source: socratic, berkeley (Question-3D), stackoverflow

So your approach 2 will be wrong. Going by standard approach i.e equating $|A-\lambda I|$ to $0$ will give coorect answer.

we will get equation as $(1-\lambda)[-(5+\lambda)(4+\lambda) +18]=0$ expand along column 1

$(\lambda +2)(\lambda-1)(1-\lambda)=0$, so eigenvalues comes out to be $x=-2, y=1, z=1$

$xy+yz+zx=-2+1-2 \Rightarrow-3$