Question

3 -1 If two eigen values of the matrix A = -1 5 -1 1 -1 3 1 - are 3 and 6, then the third eigen value is ​

Answer 1

The eigen values are -1,1,3

Explanation:

• Given Matrix A is        $\left[\begin{array}{ccc}2&1&1\\2&3&4\\-1&-1&-2\end{array}\right]$
• Therefore we use the Formula  $|A-\lambda I| = 0$

Solution:

$\left[\begin{array}{ccc}2-\lambda&1&1\\2&3-\lambda&4\\0-1&-1&-2-\lambda\end{array}\right] =0$

$\therefore (2-\lambda)[(3-\lambda)(-2-\lambda)+4]-1[-4-2\lambda+4]+[-2+3-\lambda]=0\\\\\therefore (2-\lambda)[\lambda^2 -\lambda-2]+\lambda+1=0\\\\\therefore \lambda = (-1), 1, 3$

Therefore, Eigenvalues of matrix A are −1,1,3