Properties of Determinants

The properties of determinants simplify the process of finding their values with minimal calculations. These properties are based on the elements, as well as the row and column operations, making it easier to evaluate determinants efficiently.

The properties of determinants simplify the process of their computation. Take a look at some of these properties below:

• Interchanging two rows or columns:If two rows or two columns are interchanged, the sign of the determinant changes.
  
  \( A = \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = (1 \times 4) - (2 \times 3) = -2 \)
  
  If we interchange rows,
  
  \( \begin{vmatrix} 3 & 4 \\ 1 & 2 \end{vmatrix} = (3 \times 2) - (4 \times 1) = 2 \)
  
  The sign changes from −2 to +2.
  
   
• Identical rows or columns: If any two rows or two columns are identical, the determinant equals zero. For example, 
  
  \( \begin{vmatrix} 2 & 3 \\ 2 & 3 \end{vmatrix} = (2 \times 3) - (3 \times 2) = 0 \)
  
   
• Multiplying a row or column by a constant: If each element of a row (or column) is multiplied by \(k\), the determinant is also multiplied by \(k\)
  
  \( A = \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = -2 \)
  
  Multiply first row by 2:
  
  \( \begin{vmatrix} 2 & 4 \\ 3 & 4 \end{vmatrix} = (2 \times 4) - (4 \times 3) = -4 = 2 \times (-2) \)
  
  The determinant is also multiplied by 2.
  
   
• Addition of rows or columns: If a multiple of one row (or column) is added to another, the determinant remains unchanged. For example, 
  
  \( A = \begin{vmatrix} 1 & 3 \\ 2 & 4 \end{vmatrix} = -2 \)
  
  Replace \(𝑅_2 \) by \(R_2​−3R_1\):
  
  \( \begin{vmatrix} 1 & 0 \\ 2 & -2 \end{vmatrix} = (1 \times (-2)) - (2 \times 0) = -2 \)
  
  The value remains -2.
  
   
• Determinant of a transpose: The determinant of a matrix remains the same when it is transposed. In other words,
  
  \(det(A)=det(A^T)\)
  
  For example, Original matrix:
  
  \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad \det(A) = (1 \times 4) - (2 \times 3) = -2 \)
  
  Transpose of the matrix:
  
  \( A^T = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}, \quad \det(A^T) = (1 \times 4) - (3 \times 2) = -2 \)
  
  The determinant does not change.
  
   
• Zero row or column: If all elements of a row or column are zero, the determinant is zero
  
  \( \begin{vmatrix} 0 & 0 \\ 3 & 4 \end{vmatrix} = (0 \times 4) - (0 \times 3) = 0 \)
  
   
• Factor common to a row or column: If each element of a row or column has a common factor 𝑘, it can be taken out of the determinant. For example, 
  
  \( \begin{vmatrix} 2 & 4 \\ 6 & 8 \end{vmatrix} = 2 \ \ \begin{vmatrix} 1 & 2 \\ 3 & 4 \end{vmatrix} = 2 \times (-2) = -4 \)
  
   
• Triangular matrix determinant: The determinant of a triangular matrix (upper or lower) equals the product of its diagonal elements. For example,
  
  \(\ \begin{vmatrix} 2 & 3 & 4 \\ 0 & 5 & 6 \\ 0 & 0 & 7 \end{vmatrix} = 2 \times 5 \times 7 = 70 \ \)

Mastering determinant properties helps simplify calculations, reduce errors, and solve problems faster. Practicing strategically builds confidence and accuracy.

• Start practicing with 2×2 and 3×3 matrices to understand the effects of each property before moving to larger ones.
  
   
• Simplify determinants by using row and column operations like swapping, adding, or multiplying rows/columns.
  
   
• Remember key rules such as identical rows or columns giving a determinant of zero and triangular matrices having a determinant equal to the product of diagonal elements.
  
   
• Always track the sign change when swapping rows or columns to avoid mistakes.
  
   
• Solve multiple problems applying different properties together to build speed and confidence.

Determinants are widely used in science, engineering, and mathematics to solve equations, analyze structures, and perform transformations. Understanding their properties helps apply them efficiently in real-world problems.

• Solving systems of linear equations: Determinants are used in Cramer’s Rule to find solutions for multiple equations in engineering, physics, and economics.
  
   
• Computer graphics and transformations: Determinants help calculate scaling, rotation, and reflection effects in 2D and 3D graphics.
  
   
• Engineering and structural analysis: Used to analyze stability of structures, forces in trusses, and mechanical systems.
  
   
• Cryptography: Determinants play a role in encoding and decoding messages using matrix-based encryption techniques.
  
   
• Physics and mechanics: Determinants are applied in calculating areas, volumes, and solving problems related to motion and forces.