Determinant Of matrices

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)

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

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.

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) 3 2 −1 1 b) 4 6 2 3 c) 5 0 7 −3 𝑎) 3 2 −1 1 =3− −2 =5 non-singular 𝑏) 4 6 2 3 =12− 12 =0 singular 𝑐) 3 2 −1 1 =−15− 0 =−15 non-singular

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) 4 𝑝+2 −1 3−𝑝 b) 2𝑞+1 4𝑞 2 3 c) 𝑘+10 2𝑘 −2𝑘 −2 𝑎) 4 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

WB 3ab Determinant of a 3by3 Find each determinant and decide if the matric is singular a) 2 0 5 −3 1 −5 0 2 4 b) −2 −1 1 1 −2 3 4 1 2 c) 1 2 4 3 2 1 −1 4 3 𝑎) 2 0 5 −3 1 −5 0 2 4 =2 1 −5 2 4 −0 −3 −5 0 4 +5 −3 1 0 2 =2(14)−0(−12)+5(−6) =28−0+(−30) =−2 non-singular 𝑏) −2 −1 1 1 −2 3 4 1 2 =−2 −2 3 1 2 −(−1) 1 3 4 2 +1 1 −2 4 1 =−2(−7)+1(−10)+1(9) =14−10+9 =13 non-singular

WB 3c Determinant of a 3by3 Find each determinant and decide if the matrix is singular, a) 2 0 5 −3 1 −5 0 2 4 b) −2 −1 1 1 −2 3 4 1 2 c) 1 2 4 3 2 1 −1 4 3 𝑐) 1 2 4 3 2 1 −1 4 3 =1 2 1 4 3 −2 3 1 −1 3 +4 3 2 −1 4 =1(2)−2(10)+4(14) =2−20+56 =38 non-singular

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 −2 1 2 5 0 𝑘+3 b) 𝑘 −3 4 2 2 𝑘 𝑘+3 5 5 c) 𝑘 5 𝑘+1 −𝑘 0 4 3 10 8 𝑎) 3 𝑘 0 −2 1 2 5 0 𝑘+3 =3 1 2 0 𝑘+3 −𝑘 −2 2 5 𝑘+3 +0 −2 1 5 0 =3 𝑘+3 −𝑘 −2𝑘−6 + 0 =2 𝑘 2 +19𝑘+9 If singular 2 𝑘 2 +19𝑘+9=0 2𝑘+1 𝑘+9 =0 𝑘=− 1 2 𝑘=−9

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 −2 1 2 5 0 𝑘+3 b) 𝑘 −3 4 2 2 𝑘 𝑘+3 5 5 c) 𝑘 5 𝑘+1 −𝑘 0 4 3 10 8 𝑏) 𝑘 −3 4 2 2 𝑘 𝑘+3 5 5 =𝑘 2 𝑘 5 5 +3 2 𝑘 𝑘+3 5 +4 2 2 𝑘+3 5 =𝑘 10−5𝑘 +3 10− 𝑘 2 −3𝑘 +4 10−2𝑘−6 =−8 𝑘 2 −7𝑘+46 If singular 8 𝑘 2 +7𝑘−46=0 8𝑘+23 𝑘−2 =0 𝑘=2 𝑘=− 23 8

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 −2 1 2 5 0 𝑘+3 b) 𝑘 −3 4 2 2 𝑘 𝑘+3 5 5 c) 𝑘 5 𝑘+1 −𝑘 0 4 3 10 8 𝑐) 𝑘 5 𝑘+1 −𝑘 0 4 3 10 8 =𝑘 0 4 10 8 −5 −𝑘 4 3 8 +(𝑘+1) −𝑘 0 3 10 =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=2 𝑘=−3

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 –

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