Question Number 41305 by sameer12 last updated on 05/Aug/18
![[(1,2,3),(3,4,5),(5,6,7) ] find determinant of given matrix?](https://www.tinkutara.com/question/Q41305.png)
$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{3}}\\{\mathrm{3}}&{\mathrm{4}}&{\mathrm{5}}\\{\mathrm{5}}&{\mathrm{6}}&{\mathrm{7}}\end{bmatrix} \\ $$$${find}\:{determinant}\:{of}\:{given}\:{matrix}? \\ $$
Commented by MrW3 last updated on 05/Aug/18
![row 3 −row 2: [(1,2,3),(3,4,5),(2,2,2) ] row 2−row 1: [(1,2,3),(1,1,1),(2,2,2) ] row 3−2×row 2: [(1,2,3),(1,1,1),(0,0,0) ] ⇒determinant =0](https://www.tinkutara.com/question/Q41307.png)
$${row}\:\mathrm{3}\:−{row}\:\mathrm{2}: \\ $$$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{3}}\\{\mathrm{3}}&{\mathrm{4}}&{\mathrm{5}}\\{\mathrm{2}}&{\mathrm{2}}&{\mathrm{2}}\end{bmatrix} \\ $$$${row}\:\mathrm{2}−{row}\:\mathrm{1}: \\ $$$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{3}}\\{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{2}}&{\mathrm{2}}&{\mathrm{2}}\end{bmatrix} \\ $$$${row}\:\mathrm{3}−\mathrm{2}×{row}\:\mathrm{2}: \\ $$$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{2}}&{\mathrm{3}}\\{\mathrm{1}}&{\mathrm{1}}&{\mathrm{1}}\\{\mathrm{0}}&{\mathrm{0}}&{\mathrm{0}}\end{bmatrix} \\ $$$$\Rightarrow{determinant}\:=\mathrm{0} \\ $$
Answered by malwaan last updated on 05/Aug/18
$$\left(\mathrm{1}×\mathrm{4}×\mathrm{7}+\mathrm{2}×\mathrm{5}×\mathrm{5}+\mathrm{3}×\mathrm{3}×\mathrm{6}\right)− \\ $$$$\left(\mathrm{3}×\mathrm{4}×\mathrm{5}+\mathrm{2}×\mathrm{3}×\mathrm{7}+\mathrm{5}×\mathrm{6}×\mathrm{1}\right) \\ $$$$=\left(\mathrm{28}+\mathrm{50}+\mathrm{54}\right)−\left(\mathrm{60}+\mathrm{42}+\mathrm{30}\right) \\ $$$$=\mathrm{132}−\mathrm{132}=\mathrm{0} \\ $$