Properties of Determinants : simplifying without expanding

Who is this post for?

If you only use matrices for applied calculations — for example, in linear regression using software — there's no need to memorize these properties. But if you’re in graduate school, studying linear algebra theory, or calculating determinants by hand, these properties are essential.

1st Property – Swapping Rows (or Columns)

Swapping two rows (or columns) changes the sign of the determinant.

Example: 

    Swapping the rows:

 

    2nd Property – Multiplying a Row (or Column) by a Scalar

If you multiply a row by k, the determinant is multiplied by k.

Example:

                             

Multiplying the second row by 5:

                  

    3rd Property – Null Row or Column

If a row or a column is composed only of zeros, the determinant is zero.

Example:

                            

4th Property – Equal Rows (or Columns)

If two rows or two columns are equal, the determinant is zero.

Example:

                            

5th Property – Proportional Rows (or Columns)

If one row is an exact multiple of another, the determinant is zero.

Example: 

                           

   6th Property – Triangular Matrix

If the matrix is triangular (upper or lower), the determinant is the product of the elements on the main diagonal.

Example:

                         

7th Property – Matrix Transpose

The determinant does not change when taking the transpose of a matrix.

Example: 

Conclusion

These properties are not just theoretical. They help to:

     🔹 Detect zero determinants

     🔹 Simplify matrices before expansion

     🔹 Check results efficiently

📘 In the next post: Matrix operations — how to add, subtract, and multiply.