## Key Concepts

- The determinant for [latex]\left[\begin{array}{cc}a& b\\ c& d\end{array}\right][/latex] is [latex]ad-bc[/latex].
- Cramer’s Rule replaces a variable column with the constant column. Solutions are [latex]x=\frac{{D}_{x}}{D},y=\frac{{D}_{y}}{D}[/latex].
- 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: [latex]x=\frac{{D}_{x}}{D},y=\frac{{D}_{y}}{D},z=\frac{{D}_{z}}{D}[/latex].
- 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 [latex]{A}^{-1}[/latex] is the reciprocal of the determinant of the matrix [latex]A[/latex].
- 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