Determinant of a Matrix

This article was originally part of my Numerical Linear Algebra series but since I deterred from the course of pursuing Numerical Linear Algebra and instead study Linear Algebra from a different source, I am publishing this as a standalone article. You can also read another article from this series mentioned below.

Matrix & Matrix Operations

Every square matrix of the order n x n can be associated with an expression or a number which is known as the determinant of the matrix denoted by det(A) or |A|. A matrix that is not a square matrix can not posses a determinant.

Demonstration of Determinant of a matrix A. (pardon my penmanship)

Properties of the Determinant

1. Determinant of any matrix A and its transpose Aᵀ are same, i.e., |A| = |Aᵀ|.
2. If a matrix A has a row (or column) of zeros then |A| = 0.
3. If a matrix A has two identical rows (or columns) then |A| = 0.
4. If a matrix A is triangular, i.e. it is either upper triangular matrix or lower triangular matrix, then the determinant of the matrix A will be a product of its diagonal elements.
5. If two rows (or columns) of a matrix A are interchanged the determinant of the new matrix B will be of opposite sign of determinant of matrix A, i.e., |B| = -|A|.
6. If a row (or column) of a matrix A is multiplied with a scalar k, then the determinant of the new matrix B will be k times det A, i.e., |B| = k |A|.
7. |AB| = |A||B| and |Aⁿ| = |A|ⁿ
8. For a square matrix A, if |A| ≠ 0, the inverse of A exists and |A⁻¹| = 1 / |A|.
9. Adding a scalar multiple of one row (or column) to another row (or column) does not change the value of the determinant.

Minor and Cofactors

Let A be a square matrix of order n and Mᵢⱼ is the matrix obtained by deleting iᵗʰ row and jᵗʰ column from matrix A then the determinant |Mᵢⱼ| is called the minor of the aᵢⱼ element of A. This is how a minor is defined. Furthermore, cofactor Cᵢⱼ of aᵢⱼ element of A is the signed minor Mᵢⱼ defined as Cᵢⱼ = (-1)ᶦ⁺ʲ |Mᵢⱼ|.

Example of Minor and Cofactor. (pardon my penmanship)

Determinant of matrix A can be written in the form of cofactors too.

Determinant of A in the form of cofactors. (pardon my penmanship)

Adjoint of a Matrix

As discussed in the previous article, adjoint of a matrix A is defined as the transpose of the cofactor matrix C, where C is the matrix formed by the cofactors of the elements of A, i.e., adj(A) = [Cᵢⱼ]ᵀ.

It is a well known fact that A(adj A) = (adj A)A = |A|I. Therefore, for a non-singular matrix, i.e. |A|≠0, inverse of A is defined as A⁻¹ = |A|⁻¹ adj(A).

Illustration of inverse of a matrix. (pardon my penmanship)

Again, I would like to mention the fact that inverse of a matrix is the property of square matrices only.

Crammer’s Rule

Let us now look into one of the most famous application of determinant of matrix -Crammer’s Rule.

A system of n linear equations in n unknowns is called a Crammer system if and only if the matrix A formed by the coefficients of the unknowns in the system of linear equations is non-singular, i.e., |A|≠0. A special method to solve such a system of linear equations is Crammer’s rule.

Consider a system of linear equation, a₁₁x + a₁₂y + a₁₃z = b₁, a₂₁x + a₂₂y + a₂₃z = b₂, and a₃₁x + a₃₂y + a₃₃z = b₃. Then this system of linear equations can be written as:

System of Linear Equations (pardon my penmanship)

here A = (aᵢⱼ)₃ₓ₃ , X = [x y z]ᵀ, and B = [b₁ b₂ b₃]ᵀ. Thus, AX = B. So now, if |A|≠0, the above system is Crammer’s system and it can be solved using the Crammer’s rule which is shown below.

Crammer’s Rule (pardon my penmanship)

Matrix and its operations lay the very foundations of Linear Algebra and are very crucial in Engineering and Mathematics.

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