2

How to find the possible square roots of the two rowed unit matrix I ?

I took a matrix like this $$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ and wrote,$A.A=I$

and got a set of equations

$a^2+bc=1$

$ab+bd=0$

$ac+cd=0$

$cb+d^2=1$

After that what to do?

  • 4
    Say, $b(a+d)=c(a+d)=0$ hence either $a+d=0$, or $b=c=0$. If $a+d=0$ then... If $b=c=0$ then... – Did Jan 09 '16 at 08:29

1 Answers1

1

(Elaborating on Did's comment:) You already got that for $$A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$$ $A^2 = I$ is equivalent to the four equations $$ \begin{aligned} a^2+bc &=1 \\ (a+d)b &=0 \\ (a+d)c &= 0 \\ d^2 + bc &= 1 \end{aligned} $$ Now distinguish two cases:

Case 1: $a+d \ne 0$. Then $b=c=0$ must hold. It follows that $a^2 = d^2 =1$ and therefore $a = d = \pm 1$. The solutions in this case are $$ A = I \text{ or } A = -I \, . $$ Case 2: $a+d = 0$. Then $a^2 + bc=1$ must hold, i.e. the solutions are $$ A=\begin{pmatrix} a & b \\ c & -a \end{pmatrix} \text{ for any $a, b, c$ such that $a^2 + bc=1$} \, . $$

Martin R
  • 113,040