Question

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

Answer 1

Answer on Question 43170, Math, Matrix | Tensor Analysis

$$X = \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) , \quad X^{2} = \left( \begin{array}{cc} a^{2} + b c & a b + b d \\ c a + d c & c b + d^{2} \end{array} \right) .$$

Thus, $X^{2} + 2X = \begin{pmatrix} a^{2} + 2a + bc & ab + 2b + bd \\ ca + dc + 2c & cb + d^{2} + 2d \end{pmatrix} = -5\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} .$

This is a system of four non-linear equations for $a, b, c, d$ .

Let us rewrite equations, which arise from diagonal elements in forms: $(a + 1)^2 + 4 + bc = 0$ and $(d + 1)^2 + 4 + bc = 0$ . Subtracting one equation from another, obtain $(a + 1)^2 = (d + 1)^2$ , or $d = a$ . Plugging in $d = a$ into $ca + dc + 2c = 0$ gives $d = -1$ . Hence, $a = d = -1$ .

Substituting $a = -1$ into equation $a^2 + 2a + bc = -5$ , obtain $bc = -4$ , or $c = \frac{-4}{b}$ .

Thus, $a = d = -1$ , $c = \frac{-4}{b}$ .

$$X = \left( \begin{array}{cc} -1 & b \\ \frac{-4}{b} & -1 \end{array} \right) .$$