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

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Sonia
Sonia 2014-6-6
回答： Roger Stafford 2014-6-6
Find a 2x2 matrix X=(abcd) with real entries such that X^2 + 2X = -5I, where I is a 2x2 identity matrix.

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Roger Stafford
Roger Stafford 2014-6-6
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.

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