Find a 2x2 matrix X=(abcd) with real entries such that X^2 + 2X = -5I

Find a 2x2 matrix X=(abcd) with real entries such that X^2 + 2X = -5I, where I is a 2x2 identity matrix.

[Cedric]

What have you done/tried so far?

Answer 1

[Roger Stafford]

Equations of the kind you have written can have a surprisingly large number of solutions. In the example you give, if a is any real number, b is any non-zero real number, and c and d satisfy

 c = -(a^2+3*2+5)/b
 d = -(a+2)

then

 X = [a b;c d]

always satisfies the equation X^2+2*X+5*eye(2) = 0. Thus this equation has a two-dimensional infinite continuum of possible solutions.