Factorising Determinants
Factorising Determinants
Basics
- A determinant is a special numerical value calculated from a square matrix.
- Factorising is the process of breaking up a determinant into its simplest multipliers, also called factors.
- Factoring determinants is valuable when working with matrices in more complex operations like finding inverses or evaluating systems of equations.
Determining the Determinant
- The determinant of a 2x2 matrix is found using the rule ad - bc, where a, b, c and d are the elements of the matrix.
- For larger matrices, minor determinants, also called submatrices, are created and solved.
- Determinant of a 3x3 matrix can be computed by using the “cofactor expansion” or “expansion by minors.”
- The determinant of a matrix often helps denote its properties. For example, a determinant of zero indicates a non-invertible (singular) matrix.
Factorising the Determinant
- Factorising the determinant involves breaking down the determinant into smaller factors.
- To factorise the determinant, look for common factors in the rows or columns of the matrix.
- Another effective strategy is to expand the determinant as far as possible, simplifying each term to see if any factors become apparent.
- If all of the entries of a matrix are integers, its determinant will also be an integer. Hence, the determinant can be factored into prime factors.
- Prime factorisation can be used to compare and characterise determinants of different matrices.
Applications of Factorising Determinants
- The factorised form of a determinant can provide valuable insights about the solutions of linear systems.
- Factorising determinants helps in performing matrix operations such as computing inverses and dealing with eigenvalues.
- It is particularly useful in the field of linear algebra, where complex systems require simplification for easier understanding and manipulation.
- In addition to mathematics, factorising determinants is useful in physics, computer science and economics - for example, in understanding transformation properties, performing algorithms, and analysing input-output modelling.