Question Number 33425 by Rio Mike last updated on 16/Apr/18
$${find}\:{k}\:{if}\:{the}\:{deteminant}\:{of}\: \\ $$$$\:\:\:\begin{pmatrix}{\mathrm{3}\:\:\:\:\:\:\:\:\:{k}}\\{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\mathrm{3}}\end{pmatrix}\:\:\:{is}\:\mathrm{2}\: \\ $$$${can}\:{someone}\:{please}\:{teach}\:{me}\:{how} \\ $$$${to}\:{find}\:{the}\:{deteminant}\:{of}\:{a}\:\mathrm{3}×\mathrm{3} \\ $$$${matrix}\:? \\ $$$$\: \\ $$$$\:\: \\ $$
Commented by Joel578 last updated on 16/Apr/18
$$\mathrm{9}\:−\:\mathrm{2}{k}\:=\:\mathrm{2} \\ $$$${k}\:=\:\frac{\mathrm{7}}{\mathrm{2}} \\ $$
Commented by Joel578 last updated on 16/Apr/18
$$\mathrm{For}\:\mathrm{3}×\mathrm{3}\:\mathrm{matrice},\:\mathrm{you}\:\mathrm{can}\:\mathrm{learn}\:\mathrm{from}\:\mathrm{youtube}\:\mathrm{or} \\ $$$$\mathrm{textbooks} \\ $$
Commented by prof Abdo imad last updated on 17/Apr/18
$${if}\:{A}\:=\begin{pmatrix}{{a}\:\:\:\:{b}\:\:\:\:\:{c}}\\{{e}\:\:\:\:\:{f}\:\:\:\:\:{g}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({k}\:\:\:\:{l}\:\:\:\:\:\:{m}\right. \\ $$$${det}\:{A}\:={a}\:\mid_{{l}\:\:\:\:\:\:\:\:\:{m}} ^{{f}\:\:\:\:\:\:\:{g}} \mid\:−{e}\mid_{{l}\:\:\:\:\:\:\:{m}} ^{{b}\:\:\:\:\:\:{c}} \mid\:+{k}\:\:\mid_{{f}\:\:\:\:\:\:{g}} ^{{b}\:\:\:\:\:{c}} \mid \\ $$$$={a}\left(\:{fm}−{lg}\right)\:−{e}\left({bm}−{lc}\right)\:+{k}\left({bg}\:−{fc}\right)\:{let}\:{take} \\ $$$${example} \\ $$$${A}\:=\:\begin{pmatrix}{\mathrm{1}\:\:\:\:\:−\mathrm{1}\:\:\:\:\:\:{o}}\\{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\mathrm{3}}\end{pmatrix} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\left(−\mathrm{1}\:\:\:\:\:\:\mathrm{2}\:\:\:\:\:\:\:\:\mathrm{4}\:\right) \\ $$$${det}\:{A}\:=\:\mathrm{1}\mid_{\mathrm{2}\:\:\:\:\:\:\:\mathrm{4}} ^{\mathrm{1}\:\:\:\:\:\:\:\mathrm{3}} \mid\:−\mathrm{2}\mid_{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{4}} ^{−\mathrm{1}\:\:\:\:\:\:\mathrm{0}} \mid\:+\left(−\mathrm{1}\right)\mid_{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}} ^{−\mathrm{1}\:\:\:\:\:\:\mathrm{0}} \mid \\ $$$$=\mathrm{4}−\mathrm{6}\:\:−\mathrm{2}\left(−\mathrm{4}\right)\:−\mathrm{1}\left(−\mathrm{3}\right) \\ $$$$=−\mathrm{2}\:+\mathrm{8}\:+\mathrm{3}\:=\:\mathrm{9}\:. \\ $$
Answered by MJS last updated on 16/Apr/18
$${det}\begin{pmatrix}{{a}}&{{b}}\\{{c}}&{{d}}\end{pmatrix}={ad}−{bc} \\ $$$${det}\begin{pmatrix}{{a}}&{{b}}&{{c}}\\{{d}}&{{e}}&{{f}}\\{{g}}&{{h}}&{{i}}\end{pmatrix}=_{−\left({afh}+{bdi}+{ceg}\right)} ^{\left({aei}+{bfg}+{cdh}\right)−} \\ $$