Let A 2 a 1 4 For which of the values of the constant a emb

Let A = [2, a; 1, 4].

For which of the values of the constant a embedded in matrix A will the eigenvalues of matrix A be entirely real?
Hint: Use the quadratic formula.

Solution

A = [2, a; 1, 4].

Characteristic equation: [A -Ix] =0

A -Ix = [2-x, a; 1, 4-x].

det[A-Ix] = (2-x)(4-x) -a=0

x^2 +8 -2x -4x -a =0

x^2 -6x + 8-a =0

quadratic eqution : for real solutions

b^2 -4ac>=0

36 -4*1(8-a) >=0

36 -4(8-a) >=0

9 -(8-a) >=0

1 +a>=0

a>= -1

So a =-1, 0 , 100 satisfy