11.8 Solving systems with cramer's rule  (Page 4/11)

Understanding properties of determinants

There are many properties of determinants . Listed here are some properties that may be helpful in calculating the determinant of a matrix.

Key concepts

• The determinant for $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $ad-bc.$ See [link] .
• Cramer’s Rule replaces a variable column with the constant column. Solutions are $x=\frac{D_x}{D},y=\frac{D_y}{D}.$ See [link] .
• 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). See [link] .
• 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: $x=\frac{D_x}{D},y=\frac{D_y}{D},z=\frac{D_z}{D}.$ See [link] .
• Cramer’s Rule is also useful for finding the solution of a system of equations with no solution or infinite solutions. See [link] and [link] .
• 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. See [link] and [link] .