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The determinant of a matrix is a scalar value (a single number) that can be calculated from the elements of a square matrix. The concept of a determinant is very important in linear algebra, one of its uses being to help solve systems of linear equations. Every square matrix has a unique determinant value.
A order matrix is a matrix that has two rows and two columns. Suppose we have matrix as follows:
The determinant of matrix , usually written as or , is calculated by subtracting the product of the elements of the main diagonal from the product of the elements of the second diagonal.
The formula is:
Note that the notation uses straight lines, which denote the determinant, as opposed to square brackets which denote the matrix itself.
Suppose we have matrix :
To calculate its determinant, we identify , , , and .
Then, the determinant of matrix is:
So, the determinant value of matrix is .
One important application of determinants is to solve systems of linear equations. This method is often called Cramer's Rule.
Consider the following system of linear equations with two variables (SPLDV):
In this system, and are the variables whose values we want to find. The coefficients and constants are known numbers.
This system of equations can be converted into matrix multiplication form:
The first step is to calculate the determinant of the coefficient matrix, which we call :
A system of linear equations will have a unique solution if and only if .
Next, we calculate two other determinants:
, which is the determinant of the coefficient matrix where the first column (coefficients of ) is replaced by the constant column ( ):
, which is the determinant of the coefficient matrix where the second column (coefficients of ) is replaced by the constant column ( ):
After obtaining the values of , , and , we can find the values of and using the formulas:
These formulas are only valid if .
Determine the solution of the following system of linear equations:
From the system above, we get:
, ,
, ,
Step : Calculate the determinant .
Since , this system has a unique solution.
Step : Calculate the determinant .
Step : Calculate the determinant .
Step : Calculate the values of and .
So, the solution set of the system of linear equations is and , or can be written as the ordered pair .
Given matrix and . Determine the value of .
Determine the solution of the following system of linear equations:
For matrix , its determinant is:
Given , then:
So, the value of is .
System of linear equations:
We determine and .
Then, the values of and are:
So, the solution of the system of linear equations is and , or the ordered pair .
