The determinant of a square matrix is a value calculated from the entries (or numbers) of the matrix. This value provides information about the inverse of the matrix and about the system of linear equations associated with the matrix. The determinant of matrix is denoted . If the value of the determinant of a matrix is zero, then the inverse of the matrix does not exist.The determinant of a matrix can be found using a formula.
|Matrix||Determinant||Does the inverse exist?|
|does not exist.|
Using Determinants to Solve Systems of Equations
Another method of solving systems of equations by using matrices is called Cramer's rule. To apply Cramer's rule, the system of linear equations should be in standard form.1. Let represent the matrix of coefficients of the system and represent the matrix of constants. Note that has only one column.
2. Write a new matrix that replaces one column of with the constant matrix . Write by replacing the coefficients of with the constant matrix . Write by replacing the coefficients of with the constant matrix . If there are three variables, then write by replacing the coefficients of with the constant matrix .
3. Evaluate the determinants of and the new matrices. Divide the determinants to solve for each variable.