1. Determinant
Of matrices

2. Matrices: Determinant and Inverses
KUS objectives
BAT find the Determinant of 2by2 and 3by3 matrices
Starter: 𝒂 𝒃 𝒄 𝒅 𝒆 𝒇 𝒈 𝒉 𝒊 = 𝟐 𝟎 −𝟑 𝟏 −𝟐 𝟎 𝟑 −𝟏 𝟒 Evaluate:
𝒆(𝒂𝒊−𝒄𝒈)
𝒂 𝒆𝒊−𝒇𝒉
𝒂 𝒆𝒊−𝒇𝒉 −𝒃 𝒅𝒊−𝒇𝒈 +𝒄(𝒅𝒉−𝒆𝒈)

3. Notes Determinant of a 2by2
One way to find the determinant of a 2x2 matrix is the formula below.
𝑎 𝑏 𝑐 𝑑 =𝑎𝑑 −𝑏𝑐

4. One way to find the determinant of a 3x3 matrix is the formula below.
Notes Determinant of a 3by3
One way to find the determinant of a 3x3 matrix is the formula below.

5. a) 3 2 −1 1 b) 4 6 2 3 c) 5 0 7 −3 WB 1 Determinant of a 2by2
Find each determinant and decide if the matric is singular,
a) − b) c) −3
𝑎) −1 1 =3− −2 = non-singular
𝑏) =12− 12 = singular
𝑐) −1 1 =−15− 0 =− non-singular

6. a) 4 𝑝+2 −1 3−𝑝 b) 2𝑞+1 4𝑞 2 3 c) 𝑘+10 2𝑘 −2𝑘 −2
WB 2 Determinant of a 2by2
Given that the following matrices are singular, find k, p and q
a) 𝑝+2 −1 3−𝑝 b) 𝑞+1 4𝑞 c) 𝑘+10 2𝑘 −2𝑘 −2
𝑎) 𝑝+2 −1 3−𝑝 =4 3−𝑝 −(−1) 𝑝+2 =0
𝑝= 14 3
12−4𝑝+𝑝+2=0
𝑏) 2𝑞+1 4𝑞 2 3 =3(2𝑞+1)−2 4𝑞 =0
𝑞= 3 2
6𝑞+3−8𝑞=0
𝑐) 𝑘+10 2𝑘 −2𝑘 −2 =−2 𝑘+10 − 2𝑘 (−2𝑘)=0
4 𝑘 2 −2𝑘−20=0
𝑘=−2, 𝑘= 5 2
2 𝑘 2 −𝑘−10=0
(2𝑘−5)(𝑘+2)=0

7. WB 3ab Determinant of a 3by3
Find each determinant and decide if the matric is singular
a) −3 1 − b) −2 −1 1 1 − c) −1 4 3
𝑎) −3 1 − =2 1 − −0 −3 − −
=2(14)−0(−12)+5(−6) =28−0+(−30) =− non-singular
𝑏) −2 −1 1 1 − =−2 − −(−1) −2 4 1
=−2(−7)+1(−10)+1(9) =14−10+9 = non-singular

8. WB 3c Determinant of a 3by3
Find each determinant and decide if the matrix is singular,
a) −3 1 − b) −2 −1 1 1 − c) −1 4 3
𝑐) − = − − −1 4
=1(2)−2(10)+4(14) =2− = non-singular

9. For each matrix, given that the matrix is singular, f
WB 4a Determinant of a 3by3
For each matrix, given that the matrix is singular, f
a) Find the determinant in terms of k b) find k
a) 3 𝑘 0 − 𝑘 b) 𝑘 − 𝑘 𝑘 c) 𝑘 5 𝑘+1 −𝑘
𝑎) 𝑘 0 − 𝑘+3 = 𝑘+3 −𝑘 −2 2 5 𝑘 −
=3 𝑘+3 −𝑘 −2𝑘−6 + 0
=2 𝑘 2 +19𝑘+9
If singular 𝑘 2 +19𝑘+9=0
2𝑘+1 𝑘+9 =0
𝑘=− 𝑘=−9

10. For each matrix, given that the matrix is singular, f
WB 4b Determinant of a 3by3
For each matrix, given that the matrix is singular, f
a) Find the determinant in terms of k b) find k
a) 3 𝑘 0 − 𝑘 b) 𝑘 − 𝑘 𝑘 c) 𝑘 5 𝑘+1 −𝑘
𝑏) 𝑘 − 𝑘 𝑘 =𝑘 2 𝑘 𝑘 𝑘 𝑘+3 5
=𝑘 10−5𝑘 +3 10− 𝑘 2 −3𝑘 +4 10−2𝑘−6
=−8 𝑘 2 −7𝑘+46
If singular 𝑘 2 +7𝑘−46=0
8𝑘+23 𝑘−2 =0
𝑘= 𝑘=− 23 8

11. For each matrix, given that the matrix is singular, f
WB 4c Determinant of a 3by3
For each matrix, given that the matrix is singular, f
a) Find the determinant in terms of k b) find k
a) 3 𝑘 0 − 𝑘 b) 𝑘 − 𝑘 𝑘 c) 𝑘 5 𝑘+1 −𝑘
𝑐) 𝑘 5 𝑘+1 −𝑘 =𝑘 −5 −𝑘 (𝑘+1) −𝑘
=𝑘 −40 −5 −8𝑘−12 +(𝑘+1)(−10𝑘)
=−10 𝑘 2 −10𝑘+60
If singular 𝑘 2 +𝑘−6=0
𝑘−2 𝑘+3 =0
𝑘= 𝑘=−3

12. BAT find the inverse of 2by2 and 3by3 matrices
KUS objectives
BAT find the inverse of 2by2 and 3by3 matrices
self-assess
One thing learned is –
One thing to improve is –

13. END