Question Number 32500 by 7991 last updated on 26/Mar/18
$${proof}:\:\left(−{a}\right)\left(−{b}\right)={ab} \\ $$
Answered by Joel578 last updated on 26/Mar/18
![We know (−1).(−1) = 1 and (−1) . a = −a, a ∈ R (−a)(−b) = [(−1) . a][(−1) . b] = [(−1)(−1)] [a . b] = 1 . ab = ab](Q32505.png)
$$\mathrm{We}\:\mathrm{know}\:\left(−\mathrm{1}\right).\left(−\mathrm{1}\right)\:=\:\mathrm{1} \\ $$$$\mathrm{and}\:\left(−\mathrm{1}\right)\:.\:{a}\:=\:−{a},\:\:\:{a}\:\in\:\mathbb{R}\: \\ $$$$ \\ $$$$\left(−{a}\right)\left(−{b}\right)\:=\:\left[\left(−\mathrm{1}\right)\:.\:{a}\right]\left[\left(−\mathrm{1}\right)\:.\:{b}\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\left[\left(−\mathrm{1}\right)\left(−\mathrm{1}\right)\right]\:\left[{a}\:.\:{b}\right] \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{1}\:.\:{ab} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:{ab} \\ $$