Section 8.3 – Systems of Linear Equations - DeterminantsUsing Determinants to Solve Systems of Equations

A determinant is a value that is obtained from a square matrix. 𝑎 𝑏𝑐 𝑑

2 57 1

𝐷=|2 57 1|¿2 ∙1−5 ∙7¿2−35¿−33

Section 8.3 – Systems of Linear Equations - Determinants

𝐷𝑥𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡𝑜𝑓 𝑦 :𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡 𝑜𝑓 𝑥 :

𝐷𝑦

𝑥=𝐷𝑥

𝐷

𝑦=𝐷𝑦

𝐷

Using Determinants to Solve Systems of Equations. (pg. 579 #18)

𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 :(−1 ,2)

Section 8.3 – Systems of Linear Equations - Determinants

{ 𝑥+3 𝑦=52𝑥−3 𝑦=−8

𝐷=|1 32 −3|¿1 ∙(−3)−3∙2¿−3−6 ¿−9

𝐷𝑥=| 5 3−8 −3|¿5 ∙(−3)−3 ∙(−8)¿−15+24¿9

𝐷𝑦=|1 52 −8|¿1 ∙(−8)−5∙2¿−8−10¿−18

𝑥=𝐷𝑥

𝐷¿9−9¿−1 𝑦=

𝐷𝑦

𝐷¿−18−9 ¿2

Using Determinants to Solve Systems of Equations

Determinant of a 3 x 3 Matrix

Section 8.3 – Systems of Linear Equations - Determinants

|𝑎11 𝑎12 𝑎13

𝑎21 𝑎22 𝑎23

𝑎31 𝑎32 𝑎33|

|𝑎11 𝑎12 𝑎13

𝑎21 𝑎22 𝑎23

𝑎31 𝑎32 𝑎33||𝑎11 𝑎12 𝑎13

𝑎21 𝑎22 𝑎23

𝑎31 𝑎32 𝑎33||𝑎11 𝑎12 𝑎13

𝑎21 𝑎22 𝑎23

𝑎31 𝑎32 𝑎33|

3 x 3 DeterminantsCalculate the determinant of the given matrix.

Section 8.3 – Systems of Linear Equations - Determinants

3 4 5−4 6 31 −4 3 | 3 4 5

−4 6 31 −4 3|

𝐷=¿3| 6 3−4 3|−4|−4 3

1 3|+5|−4 61 −4|

𝐷=¿3 (6 ∙3−3 ∙ (−4 ))−4 ((−4 )∙3−3∙1)+5 ((−4 )∙(−4 )−6 ∙1)

𝐷=90+6 0+5 0𝐷=200

Using Determinants to Solve Systems of EquationsSection 8.3 – Systems of Linear Equations - Determinants

𝐷≠0

Using Determinants to Solve Systems of EquationsSection 8.3 – Systems of Linear Equations - Determinants

−2 𝑥+ 𝑦−4 𝑧=45 𝑥−2 𝑦=18𝑥−5 𝑧=17

𝐷=¿−2 (−2 ∙−5−0 ∙0)−1(5∙ (−5)−0 ∙1)+(−4)(5 ∙0−(−2) ∙1)𝐷=−3

𝐷𝑥=¿4(−2 ∙(−5)−0 ∙0)−1(18 ∙(−5)−0 ∙17)+(−4)(18 ∙0−(−2) ∙17 )𝐷𝑥=−6

𝑥=𝐷𝑥

𝐷 =−6−3

=2

Using Determinants to Solve Systems of EquationsSection 8.3 – Systems of Linear Equations - Determinants

𝐷=−3

𝐷𝑦=¿−2 (18 ∙(−5)−0 ∙17)−4 (5 ∙(−5)−0∙1)+(−4)(5 ∙17−18 ∙1)𝐷𝑦=12

𝑦=𝐷𝑦

𝐷 = 12−3

=−4

𝐷𝑧=¿−2 ((−2)∙17−18 ∙0)−1(5∙17−18 ∙1)+4 (5 ∙0−(−2)∙1)𝐷𝑧=9

𝑧=𝐷𝑧

𝐷 = 9−3

=−3

Using Determinants to Solve Systems of EquationsSection 8.3 – Systems of Linear Equations - Determinants

−2 𝑥+ 𝑦−4 𝑧=45 𝑥−2 𝑦=18𝑥−5 𝑧=17

𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 : (2 ,−4 ,−3)

Properties of DeterminantsSection 8.3 – Systems of Linear Equations - Determinants

|1 2 31 2 34 5 6| 𝐷=¿1|2 3

5 6|−2|1 34 6|+3|1 2

4 5|𝐷=¿1(2 ∙6−3 ∙5)−2 (1∙6−3 ∙4)+3 (1 ∙5−2∙4 )¿−3+12−9¿0

Properties of DeterminantsSection 8.3 – Systems of Linear Equations - Determinants

𝑅1→−2𝑟2+𝑟1

Use the properties of determinants to solve the following problems.Section 8.3 – Systems of Linear Equations - Determinants

|𝑥 𝑦 𝑧𝑢 𝑣 𝑤1 2 3 |=4

𝑃𝑔 .579 ¿44

|𝑥 𝑦 𝑧𝑢 𝑣 𝑤2 4 6 |=¿2|𝑥 𝑦 𝑧

𝑢 𝑣 𝑤1 2 3 |=¿ 2(4)=¿8

𝑃𝑔 .579¿46

| 1 2 3𝑥−𝑢 𝑦−𝑣 𝑧−𝑤𝑢 𝑣 𝑤 |=¿

(−1 ) (−1 ) 4=¿4

|1 2 3𝑥 𝑦 𝑧𝑢 𝑣 𝑤|=¿

𝑅2→𝑟2+𝑟3 𝑅1↔𝑅2

(−1)|𝑥 𝑦 𝑧1 2 3𝑢 𝑣 𝑤|=¿

𝑅2↔𝑅3

(−1)(−1)|𝑥 𝑦 𝑧𝑢 𝑣 𝑤1 2 3 |=¿