## Determinants

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. 1 2 3 4 5 6 7 The system cannot be solved using Cramer’s Rule. 8   9   10   