how much matrices of integers number A= [(a,b),(c,d) ]if A^2 =I and b=c


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Question Number 12045 by 7991 last updated on 10/Apr/17

$h o w m u c h m a t r i c e s o f i n t e g e r s n u m b e r$ $A = [\begin{matrix} a & b \\ c & d \end{matrix}] i f A^{2} = I a n d b = c$
	Answered by sma3l2996 last updated on 10/Apr/17

	$A^{2} = [\begin{matrix} a & b \\ c & d \end{matrix}] [\begin{matrix} a & b \\ c & d \end{matrix}] = [\begin{matrix} a^{2} + b c & a b + d b \\ a c + d c & b c + d^{2} \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ $a^{2} + b c = 1; b (a + d) = 0; c (a + d) = 0; b c + d^{2} = 1$ $b (a + d) = 0 \Leftrightarrow b = 0 o r a = - d$ $i f b = 0 s o a^{2} = 1 \Rightarrow a = 1 o r a = - 1 a n d d = 1 o r d = - 1$ $i f b \neq 0 s o a = - d$ $a^{2} + b^{2} = 1 \Leftrightarrow b = \sqrt{1 - a^{2}} o r b = - \sqrt{1 - a^{2}}$ $A = [\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}] o r A = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}] o r A = [\begin{matrix} - 1 & 0 \\ 0 & - 1 \end{matrix}] o r A = I$ $o r A = [\begin{matrix} a & \sqrt{1 - a^{2}} \\ \sqrt{1 - a^{2}} & - a \end{matrix}] o r A = [\begin{matrix} a & - \sqrt{1 - a^{2}} \\ - \sqrt{1 - a^{2}} & - a \end{matrix}]$
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