Cramer’s Rule
Other than Gauss-Jordan, there is a powerful technique that involves
determinants to solve a system of n-equations with m-unknowns. This technique
is known as Cramer’s Rule. This technique is restricted to systems that have
the same number of equations and unknowns.
Steps To Solving A System Using Cramer’s Rule:
1. Write the coefficient matrix of the system and the column matrix b. The
column matrix b is a column matrix composed of the values to the right of the
equal sign.
2. Compute the determinant of the coefficient matrix.
3. Foe each unknown, replace the b matrix with the corresponding column. For
example, to find the value of x1 per say, replace the 1st column of the coefficient
matrix with the b matrix.
4. Find the solution, if any, to the system. The general formula for each xi is
given by
det( Ai )
xi =
, i = 1,2,3,...., n
det( A)
where Ai is the ith column replaced by b. There is no solution to the system if
det(A) = 0.
Example: Solve the following system using Cramer’s Rule.
4 x1 + 11x 2 = 7
x1 + 3 x 2 = 2
1. Write the coefficient matrix of the system and the column matrix b. The
column matrix b is a column matrix composed of the values to the right of the
equal sign.
4 11
7 
A=
b= 

1 3 
2
2. Compute the determinant of the coefficient matrix.
4 11
det( A) = det 
 = (4)(3) − (11)(1) = 1
1 3 
3. Foe each unknown, replace the b matrix with the corresponding column. For
example, to find the value of x1 per say, replace the 1st column of the coefficient
matrix with the b matrix.
7 11
det( A1 ) = det 
 = (7)(3) − (11)(2) = 21 − 2 = −1
2 3 
4 7
det( A2 ) = det 
 = (4)(2) − (7)(1) = 8 − 7 = −1
1 2 
4. Find the solution, if any, to the system. The general formula for each xi is
given by
det( Ai )
xi =
, i = 1,2,3,...., n
det( A)
where Ai is the ith column replaced by b.
det( A1 ) − 1
det( A2 ) 1
x1 =
x2 =
=
= −1
= =1
det( A)
1
det( A) 1
Hence, the solutions to the system are x1 = -1 and x2 = 1.