The matrix A matrix has two distinct real eigenvalues if an

The matrix A = |matrix has two distinct real eigenvalues if and only if k > |

Solution

det(A - I) = 0
det
-4 - .....k
3..........1 - =

= (-4 - )(1 - ) - 3k = 0

= -(4+)(1-) -3k=0

-1 [ (4+)(1-) +3k ]= 0

(4+) (1-) + 3k = 0

4 -4 + -^2 +3k = 0

-^2 -3 +4+3k=0

-1(^2 +3 -(4+3k) ) = 0

^2 +3 -(4+3k) = 0

has two distinct real eigenvalues if
the discriminant is positive

b^2 -4ac > 0 [ if equation of form ax^2 +bx +c=0 ]

(3)^2 -4.1.-(4+3k) >0

9 +4(4+3k) > 0

9 +16 +12k > 0

12k >25

k >25/12

k > 1