In high school, we are often given simple cases such as 2 or 3 equations and 2 or 3 known numbers to be solved using the Elimination Method or Substitution Method. However, this time, at the Higher Education level, we are asked to solve these equations using Matrix Determinants or Cramer's Rule. Here are the Theorems that we have successfully presented:
Determinants and Cramer's Rule
If AX = B is a system consisting of n linear equations in n unknowns such that det(A) ≠ 0, then the system has a unique solution. This solution is:
where Aj is the matrix we obtain by replacing the entries in the jth column of A with the entries in the matrix.
For more details, consider the following example. Find the solution to the equation below using Cramer's rule.
x1 + 2x3 = 6
-3x1 + 4x2 + 6x3 = 30
-x1 -- 2x2 + 3x3 = 8Convert it first into matrix form
Since the unknown number or solution is 3, it means we form matrices A1, A2 and A3. With matrix A1 formed from matrix A by replacing the first column entries in matrix A with the values on the right side equal to ( = ) in the equation above, namely:
Then to form the matrix A2, we replace the entries of the second column of matrix A with
Likewise, to form the A3 matrix, namely replacing the entries in the third column. So that A1, A2 and A3 are obtained as below;
To calculate the determinant of matrices A, A1, A2 and A3, you can use Determinants Using Cofactors.
Based on the theorem above, we obtain:
Source:
Anton, H., 1992, Elementary Linear Algebra, Erlangga, Jakarta.