This method was introduced by the Swiss mathematician Gabriel Cramer, hence the name Cramer's Rule.
Cramer's Rule
If a system of n linear equations in n variables has a coefficient matrix A with a nonzero determinant |A|, then the system has the following solution.
x_1=|A_1|/|A|,x_2=|A_2|/|A|,...,x_n=|A_n|/|A|
The ith column of A_i is the column of constants in the system of equations. If the determinant of the coefficient matrix is zero, then the system has either no solution or infinitely many solutions.
Let's try to apply this method to find the solution of the following system of equations.
2x+ 3y= 1 & (I) 8x+ 2y= 4 & (II)
We can start by writing the coefficient matrix A of this system.
A=
2 & 3 8 & 2
Next, we need to check if the determinant of A is nonzero.
det A=2*2-3*8=-20≠0 ✓
The determinant is nonzero, so we can use Cramer's Rule! Now, let's write the matrices A_1 and A_2. In these matrices, we replace the first and second columns, respectively, of matrix A with the constants from the right-hand side of the system of equations.
A_1=
1 & 3 4 & 2
A_2=
2 & 1 8 & 4
Now, let's find the determinant of each matrix using the formula for the determinant of a 2* 2 matrix.
Matrix
Use the Determinant Formula
Calculate Determinant
A= 2 & 3 8 & 2
|A|=2* 2-3* 8
|A|=- 20
A_1= 1 & 3 4 & 2
|A_1|=1* 2-3* 4
|A_1|=- 10
A_2=
2 & 1 8 & 4
|A_2|=2* 4-8* 1
|A_2|=0
Now we can find the solutions of the system of equations using Cramer's Rule.
x= |A_1||A| & (I) y= A_2|A| & (II)
(I), (II):Substitute values
x= - 10- 20 y= 0- 20
(I), (II):Calculate quotient
x=0.5 y=0
We found the solution of the system using Cramer's Rule! Now we are ready to complete the given statement.
Cramer's Rule is a method for using determinants to solve a system of linear equations.
