## Key Concepts

• The determinant for $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $ad-bc$.
• Cramer’s Rule replaces a variable column with the constant column. Solutions are $x=\frac{{D}_{x}}{D},y=\frac{{D}_{y}}{D}$.
• To find the determinant of a 3×3 matrix, augment with the first two columns. Add the three diagonal entries (upper left to lower right) and subtract the three diagonal entries (lower left to upper right).
• To solve a system of three equations in three variables using Cramer’s Rule, replace a variable column with the constant column for each desired solution: $x=\frac{{D}_{x}}{D},y=\frac{{D}_{y}}{D},z=\frac{{D}_{z}}{D}$.
• Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions.
• Certain properties of determinants are useful for solving problems. For example:
• If the matrix is in upper triangular form, the determinant equals the product of entries down the main diagonal.
• When two rows are interchanged, the determinant changes sign.
• If either two rows or two columns are identical, the determinant equals zero.
• If a matrix contains either a row of zeros or a column of zeros, the determinant equals zero.
• The determinant of an inverse matrix ${A}^{-1}$ is the reciprocal of the determinant of the matrix $A$.
• If any row or column is multiplied by a constant, the determinant is multiplied by the same factor.

## Glossary

Cramer’s Rule
a method for solving systems of equations that have the same number of equations as variables using determinants
determinant
a number calculated using the entries of a square matrix that determines such information as whether there is a solution to a system of equations