Cramer's rule - lecture -3

1. Cramer’s rule
Cramer’s rule using determinant to solve a system of Linear Equations
If 𝛥 ≠ 0 then the solutions of equations are given by
X =
𝜟𝑿
𝜟
, Y =
𝜟𝒀
𝜟
, Z =
𝜟𝒁
𝜟
Where
𝛥 is the coefficient Matrix
𝛥𝑋 is the matrix obtained from 𝛥 by replacing the coefficient of X
𝛥𝑌 is the matrix obtained from 𝛥 by replacing the coefficient of Y
𝛥𝑍 is the matrix obtained from 𝛥 by replacing the coefficient of Z

2. Q.1. Solve the following equations by using Cramer ‘s
Rule
X + Y - Z = -2
X – 2Y + Z = 3
2X - Y -3 Z = -1

3. Solution:
Step1: Write the equation in matrix form
Step 2 : Find the determinant of coefficient matrix
Step 3: Find the value of ‘X’
X =
𝜟𝑿
𝜟
Step 4: Find the value of ‘Y’
Y =
𝜟𝒀
𝜟
Step 5: Find the value of ‘Z’
Z =
𝜟𝒁
𝜟
Step 6: Check

4. X + Y - Z = -2
X – 2Y + Z = 3
2X - Y -3 Z = -1
Solution:
Step1: Write the equation in matrix form
1 1 −1
1 −2 1
2 −1 −3
𝑋
𝑌
𝑍
=
−2
3
−1

5. Step 2 : Find the determinant of coefficient matrix
𝛥 =
1 1 −1
1 −2 1
2 −1 −3
= 1( 6+1) – 1(-3-2) -1 (-1 +4)
= 7+5-3 = 9 ≠ 0
+ 1
−2 1
−1 −3
= 1(-2x -3 – (-1x1)) = 1(6+1) = 1(7) = 7
- 1
1 1
2 −3
= -1(1x- 3 – 2x1) = 1(-3 – 2) = -1(-5)= 5
+ 1
1 −2
2 −1
=-1(1x-1 – 2x-2) = -1(-1+4) =-1(3) = -3
7+5-3= 12-3= 9
As 𝛥 ≠ 0 i.e 9 we can proceed to next step.

6. Step 3: Find the value of ‘X’
X =
𝛥𝑋
𝛥
𝛥𝑋 =
−2 1 −1
3 −2 1
−1 −1 −3
𝛥𝑋 = - 2
−2 1
−1 −3
- 1
3 1
−1 −3
+(- 1)
3 −2
−1 −1
= -2 ( -2x-3- (-1x1)) – 1(3x-3 )- (-1x1) -1 (3x -1 –(-1x-2) )
= -2 ( 6+ 1) -1( -9 +1) -1(-3 -2 )
= - 2 (7) – 1 (-8) -1 (-5)
= -14+8+5
= -14+13
= -1
X =
𝛥𝑋
𝛥
=
−1
9
X = - 0.11

7. Step 4: Find the value of ‘Y’
Y =
𝛥𝑌
𝛥
𝛥𝑌 =
1 −2 −1
1 3 1
2 −1 −3
𝛥𝑌 = 1
3 1
−1 −3
- (-2)
1 1
2 −3
+(- 1)
1 3
2 −1
= 1 ( 3x-3- (-1x1)) + 2(1x-3 )- 2x1) -1 (1x -1 –2x3)
= 1 ( -9+ 1) +2( -3 -2) -1(-1 -6 )
= 1 (-8) +2 (-5) -1 (-7)
= - 8-10+57
= - 18+7
𝛥𝑌 = -11
Y =
𝛥𝑌
𝛥
=
−11
9
Y = -1.2

8. Step 5: Find the value of ‘Z’
Z =
𝛥𝑍
𝛥
𝛥𝑍 =
1 1 −2
1 −2 3
2 −1 −1
𝛥𝑍 = 1
−2 3
−1 −1
- 1
1 3
2 −1
+(- 2)
1 −2
2 −1
= 1 ( -2x-1- (-1x3)) -1(1x-1 )- 2x3) -2 (1x -1 –2x-2)
= 1 ( 2+ 3) -1( -1 -6) -2(-1 +4 )
= 1 (5) -1 (-7) -2 (3)
= 5+7-6
= -12 -6
𝛥𝑍 = 6
Z =
𝛥𝑍
𝛥
Z =
6
9
Z = 0.66

9. Solve the following equations by using Cramer ‘s Rule
1. X +2 Y +3Z = -6
4X +5 Y + 6Z = 15
7X +8Y +9 Z = 24
2. X +2 Y +Z = 3
2X +Y + 3Z = 5
2X +4Y +2Z = 7
3. X +2 Y +Z = 5
3X +Y + Z = 6
X +Y +4Z = 7

10. Ans: X = -0.11, Y =-1.2 , Z =0.66
Step 6: Check
Take any equation
X + Y - Z = -2
-0.11 – 1.2 – 0.66 = -2
-1.97 = -2
-2 = -2
X – 2Y + Z = 3
-0.11 – (2) 1.2 + 0.66 =3
2X - Y -3 Z = -1
2 ( -0.11) – (-1.2 –3) - (0.66 ) = - 1
Hence Proved.