The determinant is a characteristic of square matrices that can provide information about them that helps to solve problems. For a matrix M, the determinant can either be represented det ( M ) or with the entries of M in brackets   (i.e., instead of ).
Diagonals
The main diagonal of a matrix is the diagonal that runs from the upper left corner to the bottom right corner.
The secondary diagonal runs in the opposite direction.
Determinant of 2*2 Matrices
The determinant of a 2*2 matrix is the difference between the product of the entries on the main diagonal and the product of the entries on the secondary diagonal.
Examples 
Explanation 


Systems of Two Equations in Two Variables in Matrices
Systems of equations can be represented with matrices. A system of two equations in two variables is expressed with a 2*2 matrix, sometimes called a coefficient matrix.
Each equation is arranged in its own row. The first column of each row contains the coefficient of X, and the second column contains the coefficient of Y.
Cramer’s Rule
Determinants can be used to solve a system of two equations in two variables, as represented in a matrix. Cramer’s Rule states:
The solution of the system of equations
is given by
if
In other words, Cramer’s Rule only provides a solution if the determinant of the coefficient matrix is nonzero.
Examples 
Explanation 
Solve the system

The solution is .(1. 1) 
Try these exercises:
Solve
1.  Find the determinant.

2.  Find the determinant.

3.  Find the determinant.

4.  Express the system as a coefficient matrix.

5.  Express the system as a coefficient matrix.

6.  Express the system as a coefficient matrix.

7.  Determine whether or not the following system can be solved using Cramer’s Rule:

8.  Solve the system of equations.

9.  Solve the system of equations.

10.  Solve the system of equations.

Answers to questions
1.


2. 

3. 

4. 

5. 

6. 

7. 
The system cannot be solved using Cramer’s Rule.

8. 

9. 

10. 
