Question

Find the sum and product of Eigen value of matrix A = 3 1 4 0 2 6 0 0 5​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{A=\left(\begin{array}{ccc}3&1&4\\0&2&6\\0&0&5\end{array}\right)}$

$\underline{\textbf{To find:}}$

$\textsf{Sum and product of eigen values of A}$

$\underline{\textbf{Solution:}}$

$\textsf{Characteristic polynomial of A is}\;\mathsf{|A-mI|=0}$

$\mathsf{\left|\left(\begin{array}{ccc}3&1&4\\0&2&6\\0&0&5\end{array}\right)-m\left(\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right)\right|=0}$

$\mathsf{\left|\begin{array}{ccc}3-m&1&4\\0&2-m&6\\0&0&5-m\end{array}\right|=0}$

$\textsf{Expanding along first row, we get}$

$\mathsf{(3-m)[(2-m)(5-m)-0]-1[0-0]+4[0-0]=0}$

$\mathsf{(3-m)(2-m)(5-m)=0}$

$\implies\mathsf{m=2,3,5}$

$\implies\textsf{Eigen values are 2,3 and 5}$

$\mathsf{Now,}$

$\textsf{Sum of the eigen values}$

$\mathsf{=2+3+5}$

$\mathsf{=10}$

$\textsf{Product of the eigen values}$

$\mathsf{=2{\times}3{\times}5}$

$\mathsf{=30}$

Answer 2

Given:

The Eigen value of matrix A=$\left[\begin{array}{ccc}3&1&4\\0&2&6\\0&0&5\end{array}\right]$.

To find:

The Eigen value sum and product.

Step-by-step explanation:

Sum of the Eigen values= Sum of the principal diagonal elements

$=3+2+5$

$=5+5$

$=10$

product of the Eigen values=|$A$|

$=3(10-0)-1(0-0)+4(0-2)$$=30-8$

$=22$

Sum=$10$,Product=$22$

Answer:

Therefore, The Eigen value of matrix Sum=$10$ and product=$22$.