Question

What is the difference between Determinant and Matrix?

Answer 1

Here in this question, we have to explain the difference between the Determinant and Matrix. To explain or discuss first we have to learn the definition of Determinant and Matrix and next identify the similarities and the differences between by comparing both of them.

Complete answer: In mathematics especially in linear algebra, Determinants and Matrices are used to solve linear equations by applying a Cramer’s rule to a set of non-homogeneous equations which are in the linear form.
Matrix - A rectangular arrangement of $mn$ elements in the form of an ordered set of $m$ rows and $n$ columns, Suppose the number of rows is m and columns is n, then the matrix is called as $m \times n$ matrix.
The representation of $m \times n$ matrix is: A = {\left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{...}&{{a_{1n}}} \\\ {{a_{21}}}&{{a_{22}}}&{...}&{{a_{2n}}} \\\ {...}&{...}&{...}&{...} \\\ {{a_{n1}}}&{{a_{n2}}}&{...}&{{a_{nn}}} \end{array}} \right]_{m \times n}}
In compact form, the matrix is written as: $A = {\left[ {{a_{ij}}} \right]_{m \times n}}$, $1 \leqslant i < m$, $1 \leqslant j < n$.
Determinant – To every square matrix of order $A = \left[ {{a_{ij}}} \right]$ of order $n$, ${a_{ij}} \in R$, we can associate a unique real number called determinant of matrix $A$, denoted by $\det A$ or $\left| A \right|$.
The determinant of $3 \times 3$ matrix represented as: A = \left| {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}&{{a_{13}}} \\\ {{a_{21}}}&{{a_{22}}}&{{a_{23}}} \\\ {{a_{31}}}&{{a_{32}}}&{{a_{33}}} \end{array}} \right|
Difference between Matrix and Determinant:
In a matrix, the set of numbers are arranged by two rectangular brackets whereas, in a determinant, the set of numbers are covered by two bars or by modulus sign.
A matrix is a group of numbers but a determinant is a unique number related to that matrix.
In a matrix the number of rows need not be equal to the number of columns whereas, in a determinant, the number of rows should be equal to the number of columns.
Determinants are used to calculate the inverse of the matrix and if the determinant is zero then the inverse of the matrix doesn’t exist.

Note:
Remember, Determinant can be expressed only for the square matrix; it means the number of rows should be equal to the number of columns, otherwise we can’t represent determinants generally, the determinant is a scalar value and is computed from the elements of the square matrix. But The matrix can write for any number of rows and columns but write within rectangular brackets.