1. using inverse matrices
Solving
systems of equations
using inverse matrices

2. Matrices: Inverses and Systems of equations
KUS objectives
BAT solve systems of equations using inverse matrices – modelling with matrices
Starter: find the determinant of each matrix
3 2 − − 5 2
3𝑎 6𝑎 −2𝑎 −4𝑎
2𝑘 𝑘+2 𝑘 𝑘+1

3. a) −𝑥+6𝑦−2𝑧=21 6𝑥−2𝑦−𝑧=−16 −2𝑥+3𝑦+5𝑧=24
WB11a Use an inverse matrix to solve the simultaneous equations
a) −𝑥+6𝑦−2𝑧=21
6𝑥−2𝑦−𝑧=−16
−2𝑥+3𝑦+5𝑧=24
Write as a matrix multiplication
−1 6 −2 6 −2 −1 − 𝑥 𝑦 𝑧 = 21 −16 24
𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑖𝑠 −
Find the inverse matrix (using calc)
Multiply through by the inverse
𝑥 𝑦 𝑧 = − − =
− =
−1 4 2
So the solution is 𝒙=−𝟏 𝒚=𝟒 𝒛=𝟐

4. WB11b Use an inverse matrix to solve the simultaneous equations
3𝑦+3𝑧=2
−3𝑥−𝑦+3𝑧=−2
Write as a matrix multiplication
3 0 − −3 − 𝑥 𝑦 𝑧 = 0 2 −2
𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑖𝑠 (18) −12 −6 − −9 −3 −9
Find the inverse matrix (using calc)
Multiply through by the inverse
𝑥 𝑦 𝑧 = 1 (18) −12 −6 − −9 −3 − −2 = =
So the solution is 𝒙= 𝟒 𝟑 𝒚=𝟎 𝒛= 𝟐 𝟑

5. The savings account increased in value by 1.5%
WB 12 Tyler invested a total of £5000 across three different accounts; a savings account; a property bond and a share dealing account
Tyler invested £400 more in the property bond account than in the saving account
After one year
The savings account increased in value by 1.5%
The property bond account had increased in value by 3.5%
The share dealing account had decreased in value by 2.5%
The total value across Tylers three accounts had increased by £79
Form and solve a matrix equation to find out how much money was invested by Tyler in each account
𝒙 Saccount
𝒚 PB account
𝒛 SD account
Total at start is
𝒙 𝒚 𝒛=𝟓𝟎𝟎𝟎
100 more males at start
𝒙 − 𝒚 = −𝟒𝟎𝟎
Question from Specimen materials 2018
After 1 year
1.015𝒙+1.035𝒚+0.975𝒛=𝟒𝟗𝟐𝟏
− 𝑥 𝑦 𝑧 = −

6. WB12 (cont) Form and solve a matrix equation to find out how much money was invested by Tyler in each account
− 𝑥 𝑦 𝑧 = −
𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑖𝑠 − −9.75 − −0.2 −20
Find the inverse matrix (using calc)
Multiply through by the inverse
𝑥 𝑦 𝑧 = − −9.75 − −0.2 − − =
So the solution is
£𝟐𝟐𝟎 Saccount
£𝟔𝟐𝟎 PB account
£𝟒𝟏𝟔𝟎 SD account
Check this works in the original question

7. WB13 A colony of 1000 mole rats is made up of adult males, adult females and youngsters. Originally there were 100 more adult females than adult males
After one year
The number of adult males had increased by 2%
The number of adult females had increased by 3%
The number of youngsters had decreased by 4%
The total number mole-rats had decreased by 20
Form and solve a matrix equation to find out how many of each type of mole-rat were in the original colony
𝒙 adult males
𝒚 adult females
𝒛 youngsters
Total at start is
𝒙 𝒚 𝒛=𝟏𝟎𝟎𝟎
100 more males at start
𝒙 − 𝒚 = −𝟏𝟎𝟎
After 1 year
1.02𝒙+1.03𝒚+0.96𝒛=𝟗𝟖𝟎
− 𝑥 𝑦 𝑧 = −

8. WB 13 (cont) Form and solve a matrix equation to find out how many of each type of mole-rat were in the original colony
− 𝑥 𝑦 𝑧 = −
𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑖𝑠 − −96 − −1 −200
Find the inverse matrix (using calc)
Multiply through by the inverse
𝑥 𝑦 𝑧 = − −96 − −1 − − = =
So the solution is
𝟏𝟎𝟎 adult males
𝟐𝟎𝟎 adult females
𝟕𝟎𝟎 youngsters
Check this works in the original question

9. 1) If the determinant ≠ 0 then the system has a unique solution
Notes To determine if a system of equations is consistent or inconsistent
1) If the determinant ≠ 0 then the system has a unique solution
and is consistent
Three planes intersect at a single point
2) If the determinant = 0 the matrix is singular then there are two cases
i) the matrix is singular and consistent and
has infinite solutions
Three planes intersect at a line
The planes form a ‘SHEAF’
ii) the matrix is singular and inconsistent and has NO solutions
You will meet this more in the chapter on Vectors and planes
Three planes do not intersect at the same points anywhere or are parallel

10. 3𝑥−𝑘𝑦−6𝑧=𝑘 𝑘𝑥+3𝑦+3𝑧=2 −3𝑥−𝑦+3𝑧=−2
WB 15a For each of the following values of k, determine whether the system of equations below is consistent or inconsistent. If consistent, determine whether there is a unique solution or an infinity of solutions. In each case, identify the geometric configuration of the planes corresponding to each value of k a) k = b) k = c) k = -6
3𝑥−𝑘𝑦−6𝑧=𝑘
𝑘𝑥+3𝑦+3𝑧=2
−3𝑥−𝑦+3𝑧=−2
3 −𝑘 −6 𝑘 3 3 −3 − 𝑥 𝑦 𝑧 = 𝑘 2 −2
a) If k = − −3 − 𝑥 𝑦 𝑧 = 0 2 −2
3 0 − −3 −1 3 =
−1 3 −0+ − −3 −1 =3 12 −6 9 =−18
1) If the determinant ≠ 0 then the system has a unique solution
and is consistent
Three planes intersect at a single point

11. WB 15b For each of the following values of k, determine whether the system of equations below is consistent or inconsistent. If consistent, determine whether there is a unique solution or an infinity of solutions. In each case, identify the geometric configuration of the planes corresponding to each value of k a) k = b) k = c) k = -6
b) If k = −1 − −3 − 𝑥 𝑦 𝑧 = 1 2 −2
3 −1 − −3 −1 3 =
−1 3 − − − −3 −1 =3 12 −1(12)−6 8 =0
2) If the determinant = 0 the matrix is singular then there are two cases
You need to determine if the system is consistent: Eliminate one of the variables from the set of equations. If the resulting two equations are consistent then the system is consistent
3𝑥−𝑦−6𝑧= (1)
𝑥+3𝑦+3𝑧= (2)
−3𝑥−𝑦+3𝑧=−2 (3)
5𝑥+5𝑦= (1)+2x(2)
4𝑥+4𝑦= (2) – (3)
the system is consistent: equations (4) and (5) are consistent because one is a linear multiple of the other. Any values of x and y that satisfy (4) also satisfy (5)

12. WB 15b For each of the following values of k, determine whether the system of equations below is consistent or inconsistent. If consistent, determine whether there is a unique solution or an infinity of solutions. In each case, identify the geometric configuration of the planes corresponding to each value of k a) k = b) k = c) k = -6
c) If k = −6 −6 3 3 −3 − 𝑥 𝑦 𝑧 = −6 2 −2
3 6 −6 −6 3 3 −3 −1 3 =
−1 3 −6 −6 3 − −6 −6 3 −3 −1 =3 12 −6(−9)−6 15 =0
2) If the determinant = 0 the matrix is singular then there are two cases
You need to determine if the system is consistent: Eliminate one of the variables from the set of equations. If the resulting two equations are consistent then the system is consistent
3𝑥+6𝑦−6𝑧=− (1)
−6𝑥+3𝑦+3𝑧= (2)
−3𝑥−𝑦+3𝑧=− (3)
5𝑦−3𝑧=− (1)+ (3)
15𝑦−9𝑧=− x(1) + (2)
the system is Inconsistent and has no solutions: equations (4) and (5) are inconsistent because they are not solvable. No values of y and z satisfy both simultaneously

13. One thing to improve is –
KUS objectives
BAT solve systems of equations using inverse matrices – modelling with matrices
self-assess
One thing learned is –
One thing to improve is –

14. END