A is a 2 by 2 matrix det A 3 and Trace A 4 Give the small

A is a 2 by 2 matrix, det A = - 3 and Trace A = 4. Give the smallest eigenvalue of A. A is a 3 by 3 matrix. One of the eigenvalues of A is 2. Also, det A = 2 and Trace A = 5. Give the smallest eigenvalue of A

Solution

1) let the eigen values be x, y

Now product of egen values = determinant of matrix

x*y = -3 ----(1)

sum of diagonal entrioes = sum of eigen values = trace

x +y = 4 ---(2)

solve 1 and 2: x +-3/x =4

x^2 -4x -3 =0

x = ( 4 +/- sqrt(16+12) )/2 = (4 +/-2sqrt7)/2

= 2 + /- sqrt7

y = 4-x = 4 - (2+/-sqrt7) = 2 +sqrt7 , 2-sqrt7

Now from two values of x and two values of y smaalest eigen value is 2 -sqrt7

2) 3 x3 matrix , ine eigen value = 2.Let the other two eigen values be x and y

So, again : x*y*2 = 2----> xy =1 ---(1)

x +y +2 = 5 -----> x +y = 3 -----(2)

Solving equation 1 and 2:

x +1/x = 3

x^2 -3x +1 =0

x = ( 3 +/-sqrt(9+4)/2 = (3 +/- sqrt13)/2

y = 3 - (3 +/- sqrt13)/2

= 3/2 +/-sqrt13/2

So, smallest eigen value: 3/2 - sqr(13)/2