Question and Answers Forum

All Questions      Topic List

Matrices and Determinants Questions

Previous in All Question      Next in All Question      

Previous in Matrices and Determinants      Next in Matrices and Determinants      

Question Number 9285 by suci last updated on 28/Nov/16

find the determinant of the matrix below [(1,4,(−3),1),(2,0,6,3),(4,(−1),2,5),(1,0,(−2),4) ]

$${find}\:{the}\:{determinant}\:{of}\:{the}\:{matrix}\:{below} \\ $$$$\begin{bmatrix}{\mathrm{1}}&{\mathrm{4}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{2}}&{\mathrm{0}}&{\mathrm{6}}&{\mathrm{3}}\\{\mathrm{4}}&{−\mathrm{1}}&{\mathrm{2}}&{\mathrm{5}}\\{\mathrm{1}}&{\mathrm{0}}&{−\mathrm{2}}&{\mathrm{4}}\end{bmatrix} \\ $$

Answered by mrW last updated on 28/Nov/16

∣A∣= determinant ((1,4,(−3),1),(2,0,6,3),(4,(−1),2,5),(1,0,(−2),4)) R4−R1 and R3−R2×2: ∣A∣= determinant ((1,4,(−3),1),(2,0,6,3),(0,(−1),(−10),(−1)),(0,(−4),1,3)) R2−R1×2: ∣A∣= determinant ((1,4,(−3),1),(0,(−8),(12),1),(0,(−1),(−10),(−1)),(0,(−4),1,3)) =1× determinant (((−8),(12),1),((−1),(−10),(−1)),((−4),1,3)) R2+R1: =1× determinant (((−8),(12),1),((−9),2,0),((−4),1,3)) R3−R1×3: =1× determinant (((−8),(12),1),((−9),2,0),((20),(−35),0)) =1×1× determinant (((−9),2),((20),(−35))) =(−9)×(−35)−2×20=275

$$\mid\mathrm{A}\mid=\begin{vmatrix}{\mathrm{1}}&{\mathrm{4}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{2}}&{\mathrm{0}}&{\mathrm{6}}&{\mathrm{3}}\\{\mathrm{4}}&{−\mathrm{1}}&{\mathrm{2}}&{\mathrm{5}}\\{\mathrm{1}}&{\mathrm{0}}&{−\mathrm{2}}&{\mathrm{4}}\end{vmatrix} \\ $$$$\mathrm{R4}−\mathrm{R1}\:\mathrm{and}\:\mathrm{R3}−\mathrm{R2}×\mathrm{2}: \\ $$$$\mid\mathrm{A}\mid=\begin{vmatrix}{\mathrm{1}}&{\mathrm{4}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{2}}&{\mathrm{0}}&{\mathrm{6}}&{\mathrm{3}}\\{\mathrm{0}}&{−\mathrm{1}}&{−\mathrm{10}}&{−\mathrm{1}}\\{\mathrm{0}}&{−\mathrm{4}}&{\mathrm{1}}&{\mathrm{3}}\end{vmatrix} \\ $$$$\mathrm{R2}−\mathrm{R1}×\mathrm{2}: \\ $$$$\mid\mathrm{A}\mid=\begin{vmatrix}{\mathrm{1}}&{\mathrm{4}}&{−\mathrm{3}}&{\mathrm{1}}\\{\mathrm{0}}&{−\mathrm{8}}&{\mathrm{12}}&{\mathrm{1}}\\{\mathrm{0}}&{−\mathrm{1}}&{−\mathrm{10}}&{−\mathrm{1}}\\{\mathrm{0}}&{−\mathrm{4}}&{\mathrm{1}}&{\mathrm{3}}\end{vmatrix} \\ $$$$=\mathrm{1}×\begin{vmatrix}{−\mathrm{8}}&{\mathrm{12}}&{\mathrm{1}}\\{−\mathrm{1}}&{−\mathrm{10}}&{−\mathrm{1}}\\{−\mathrm{4}}&{\mathrm{1}}&{\mathrm{3}}\end{vmatrix} \\ $$$$\mathrm{R2}+\mathrm{R1}: \\ $$$$=\mathrm{1}×\begin{vmatrix}{−\mathrm{8}}&{\mathrm{12}}&{\mathrm{1}}\\{−\mathrm{9}}&{\mathrm{2}}&{\mathrm{0}}\\{−\mathrm{4}}&{\mathrm{1}}&{\mathrm{3}}\end{vmatrix} \\ $$$$\mathrm{R3}−\mathrm{R1}×\mathrm{3}: \\ $$$$=\mathrm{1}×\begin{vmatrix}{−\mathrm{8}}&{\mathrm{12}}&{\mathrm{1}}\\{−\mathrm{9}}&{\mathrm{2}}&{\mathrm{0}}\\{\mathrm{20}}&{−\mathrm{35}}&{\mathrm{0}}\end{vmatrix} \\ $$$$=\mathrm{1}×\mathrm{1}×\begin{vmatrix}{−\mathrm{9}}&{\mathrm{2}}\\{\mathrm{20}}&{−\mathrm{35}}\end{vmatrix} \\ $$$$=\left(−\mathrm{9}\right)×\left(−\mathrm{35}\right)−\mathrm{2}×\mathrm{20}=\mathrm{275} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com