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cramers rule

cramers rule


 

Grade:

1 Answers

Godfrey Classic Prince
633 Points
8 years ago

Dear romil gayo,

Cramer’s rule is a method for solving linear simultaneous equations. It makes use of determinants. 

In linear algebra, Cramer''s rule is a theorem, which gives an expression for the solution of a system of linear equations with as many equations as unknowns, valid in those cases where there is a unique solution.

Eg : 

To find rules (or formulas) that may be used solve any system of equations, we need to solve the general system of the form 

a 1 x + b 1 y = c 1    (1)
a 2 x + b 2 y = c 2    (2)

We multiply equation (1) by b 2 and equation (2) by -b 1 and add the right and left hand terms. 

b 2 a 1 x + b 2 b 1 y = b 2 c 1    (1)
- b 1 a 2 x - b 1 b 2 y = - b 1 c 2    (2)____________________________
b 2 a 1 x - b 1 a 2 x = b 2 c 1 - b 1 c 2
Assuming that a 1 b 2 - a 2 b 1 is not equal to zero, solve the above equation for x to obtain. 

x = ( c 1 b 2 - c 2 b 1 ) / ( a 1 b 2 - a 2 b 1 )
We can use similar steps to eliminate x and solve for y to obtain. 

y = ( a 1 c 2 - a 2 c 1 ) / ( a 1 b 2 - a 2 b 1 )

Using the determinant of a Matrix notation, the solution to the given 2 by 2 system of linear equations is given by 

x = D x / D and y = D y / D 

where D, D x and D y are the determinants defined by 

cramer''s rule for 2 by 2 system of linear equations.

For a 3 by 3 system of linear equations of the form 

a 1 x + b 1 y + c 1 z = d 1    (1)
a 2 x + b 2 y + c 2 z = d 2    (2)
a 3 x + b 3 y + c 3 z = d 3    (3)

Cramer''s rule gives the solution as follows 

x = D x / D , y = D y / D and z = D z / D 

where D, D x, D y and D z are determinants defined by 

cramer''s rule for 3 by 3 system of linear equations.

Hope this helped you immensely..!

All the Very Best & Good Luck to you ..!!

 

Best Regards,

AskIITians Expert,

Godfrey Classic Prince

IIT-M

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