Finding the determinant of a 3x3 matrix

Finding the Determinant of a 3x3 Matrix

Definition of Determinant

• The determinant of a square matrix gives a special value that can provide important information about the matrix.
• For a 3x3 matrix, the determinant helps us determine if the matrix is invertible or singular.

Method for 3x3 Matrix

• The determinant of a 3x3 matrix, A can be found using the formula:
  det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
• Here, a, b, c, d, e, f, g, h and i are the entries of matrix A.

Row Reduction

• One common method to find the determinant of a 3x3 matrix is row reduction.
• Row reduction involves changing rows in the matrix into a row-echelon form or fully reduced form.
• Once completed, the determinant is simply the product of the leading entries.

Cofactor Expansion

• Another way to find the determinant is through cofactor expansion, also known as Laplace’s formula.
• This involves taking each element in a row or a column, multiplying it by its corresponding minor’s determinant, and alternate subtracting and adding the results.

Tips for Success

• Be comfortable both performing row reduction and cofactor expansion methods, as different situations may call for different methods.
• Make sure to alternate between subtraction and addition in the cofactor expansion, forgetting to switch can lead to a wrong answer.
• Be careful with your calculations and check your work for errors, as a single mistake can lead to a significant difference in the resulting determinant.