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Cramer's Rule
Gabriel Cramer was a Swiss
mathematician (1704-1752)

Introduction
 Cramer’s Rule is a method for solving linear
simultaneous equations. It makes use of
determinants and so a knowledge of these is
necessary before proceeding.
 Cramer’s Rule relies on determinants

Cramer’s Rule
 Not all systems have a definite solution. If the
determinant of the coefficient matrix is zero, a
solution cannot be found using Cramer’s Rule
because of division by zero.
 When the solution cannot be determined, one of
two conditions exists:
 The planes graphed by each equation are parallel
and there are no solutions.
 The three planes share one line (like three pages of
a book share the same spine) or represent the same
plane, in which case there are infinite solutions.

Coefficient Matrices
 You can use determinants to solve a system of
linear equations.
 You use the coefficient matrix of the linear
system.
 Linear System Coeff Matrix
ax+by=e
cx+dy=f

Cramer’s Rule for 2x2 System
 Let A be the coefficient matrix
 Linear System Coeff Matrix
ax+by=e
cx+dy=f
 If detA 0, then the system has exactly one
solution:
and
= ad – bc

Key Points
 The denominator consists of the coefficients
of variables (x in the first column, and y in the
second column).
 The numerator is the same as the denominator,
with the constants replacing the coefficients
of the variable for which you are solving.

Example - Applying Cramer’s Rule
on a System of Two Equations
Solve the system:
 8x+5y= 2
 2x-4y= -10
The coefficient matrix is: and
So:
and

Applying Cramer’s Rule
on a System of Two Equations

Evaluating a 3x3 Determinant
(expanding along the top row)
 Expanding by Minors (little 2x2 determinants)

Using Cramer’s Rule
to Solve a System of Three Equations
Consider the following set of linear equations

The system of equations above can be written in
a matrix form as:

Define

where

Example 1
Consider the following equations:

Cramer’s Rule - 3 x 3
 Consider the 3 equation system below with variables
x, y and z:

Cramer’s Rule - 3 x 3
 The formulae for the values of x, y and z are
shown below. Notice that all three have the
same denominator.

Example 1
 Solve the system : 3x - 2y + z = 9
x + 2y - 2z = -5
x + y - 4z = -2

Example 1
The solution is
(1, -3, 0)

Conclusion
Cramer’s Rule is a very efficient and perfect method to
find the solutions in the matrix. Here, it is provided that
we have the same number of equations as unknowns.
This Cramer’s Rule will give us the unique solution to
a system of all the equations if it exists. Unlike normal
equations here, we don’t have to be dependent on other
variables to know the value of the third
variable. Cramer’s rule is a method to solve the
equations but in the form of a matrix, where there are
the same amount of unknowns as equations in the
system.

References
 http://www.mathcentre.ac.uk
 Fundamentals Methods of Mathematical
 Engineering Mathematics (Pal & Das)
 https://unacademy.com

RCC INSTITUTE OF INFORMATION TEC
DEPARTMENT: ELECTRONICS & COMMUNICATION ENGIN
CONTINUOUS ASSESSMENT -1(CA1)
ACADEMIC SESSION: 2023-24 (ODD SEM
• Paper Name: Mathematics - IB
• Paper Code: BS-M102
• Year & Semester: 1st year and 1st semester
• Name of the Student: xxxxxxxxx
• Roll Number: xxxxxxxx
• Registration Number: xxxxxx
PRESENTATION TITLE

 Introduction
 Topic 1
 Topic 2
 …..
 ……..
 Conclusion
 Reference

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