** Matrix vs Determinant
**

Matrices and Determinants are important concepts is Linear Algebra, where matrices provide a concise way of representing large linear equations and combination while determinants are uniquely related to a certain type of matrices.

**More about Matrix**

Matrices are rectangular arrays of numbers where the numbers are arranged in rows and columns. The number of columns and rows in a matrix determine the size of the matrix. Generally, a matrix is identically represented by square brackets, and the numbers are aligned in rows and columns inside.

A is known as a 3×3 matrix because it has 3 columns and 3 rows. The numbers denoted by a_ij are called elements and uniquely identified by the row number and column number. Also, the matrix can be represented as [a_ij ]_(3×3) , but its uses are limited since the elements are not explicitly given. Extending the above example to a general case we can define a general matrix of size m×n;

A has m rows and n columns.

Matrices are categorized based on their special properties. As an example, a matrix with an equal number of rows and columns is known as a square matrix, and a matrix with a single column is known as a vector.

Operations on matrices are specifically defined but follow the rules in abstract algebra. Therefore, the addition, subtraction, and multiplication between matrices are performed on an element wise. For matrices, the division is not defined though the inverse exists.

Matrices are a concise representation of a collection of numbers, and it can be easily used for solving linear equation. Matrices also have wide application in the field of Linear algebra, concerning linear transformations.

**More about Determinant**

The determinant is a unique number associated with each square matrix and is obtained after performing a certain calculation for the elements in the matrix. In practice, a determinant is denoted by putting a modulus sign for the elements in the matrix. Therefore, the determinant of A is given by;

and generally for a m×n matrix

The operation for obtaining the determinant is as follows;

|A| = ∑^{n}_{j=1} a_{j} C_{ij}, where C_{ij} is the cofactor of the matrix given by C_{ij }= (-1)^{i+j} M_{ij}.

The determinant is an important factor determining the properties of the matrix. If the determinant is zero for a certain matrix, the inverse of the matrix does not exist.

**What is the difference between Matrix and Determinant?**

• A matrix is a group of numbers, and a determinant is a unique number related to that matrix.

• A determinant can be obtained from square matrices, but not the other way around. A determinant cannot give a unique matrix associated with it.

• The algebra concerning the matrices and determinants has similarities and differences. Especially when performing multiplications. For example, multiplication of matrices has to be done element wise, where determinants are single numbers and follows simple multiplication.

• Determinants are used to calculate the inverse of the matrix and if the determinant is zero the inverse of the matrix does not exist.

laxman says

why you can’t evaluate matrix?

Amey Mane says

Simply because there is no rule or method to expand a matrix. Only square matrices have determinants, and so rectangular matrices cannot be expanded.