Determinants-Determinant Matrices

The origin of determinant matrices can be traced back to the 4th century BC. However, its proper usage began in the 17th century within the ancient city of Babylon. Today, determinant matrices are used as fundamentals to be applied in various studies like mathematics, physics, computer science, etc. The following article will explain matrices, their types, operations on matrices, adjoint and inverse of a matrix, and determinants and their properties. 

Matrix definition 

A matrix is nothing but a rectangular arrangement of elements in rows and columns. If there are m number of rows and n number of columns, then the matrix is of order mn. A matrix is usually enclosed within ( ) or [ ] brackets. The element’s position within a matrix is denoted by aij where i denotes the number of the row and j denotes the number of the column. For example – a11, a12 , etc. 

Types of Matrices

The various types of matrices are listed below –

Row Matrix 

A matrix having a single row is called a row matrix.  Example:

Column Matrix

A matrix having only 1 column is called a column matrix. Example:

Zero Matrix (Null matrix)

A matrix in which all elements have the value of 0. Example:

Square Matrix

Square Matrices have the same number of rows and columns. The smallest square matrix is a matrix of order 11.

Diagonal Matrix

Diagonal matrices have any non-zero or zero element on the diagonals. However, the non-diagonal elements must be 0. 

Example: and are both diagonal matrices.

Scalar Matrix

The scalar matrix is nothing but a special kind of diagonal matrix where all elements on the diagonal have equal value. 

Example: is a scalar matrix of order 3.

Unit Matrix

When a scalar matrix has all the elements on the diagonal with the value of 1, it is called a unit matrix. 

Example:  

Operations on Matrices 

Transpose of a Matrix

Transposing a matrix is similar to flipping over or turning over the entire thing. In simple terms, the matrix obtained after interchanging the rows and the columns of a matrix Z is called the transpose of Z or simply Z’ (read as Z transpose).

For example: In the matrix, its transpose will be .

Addition and subtraction of matrices

Basic mathematical operations like addition and subtraction can be performed on the matrices of the same order. For example, if A and B are two matrices of order 33,  then the calculations can be done for each element individually with the corresponding element of the next matrix. Element a11 and element b11 must be added or subtracted according to the requirements. 

Multiplication of Matrices

Multiplication of matrices is a slightly different process that requires the number of rows in matrix A to be equal to the number of columns in matrix B, then and only then is the product possible. For example: If matrix A is of order 23 and matrix B is of order 12, then the number of rows in matrix A is equal to the number of columns in matrix B, i.e 2. Hence, the multiplication is possible. 

Determinants

Determinants were first identified in Chinese mathematics around the 3rd century BC. Determinants are square matrices that have a fixed value. These are denoted by det A or |A| for any determinant A. The value of any determinant is used to find its inverse. The value of determinants for square matrices of different orders can be calculated as follows –

• For a 22 matrix, the value is nothing but the difference between the cross-product of its elements. 

• For a 33 matrix, the value can be calculated by at first selecting a row. Then we need to find the product of each element and its corresponding cofactor. This will give us 3 entries. The next step is adding all of it together.

The value of a determinant can also be calculated through the expansion of the rows and columns of a determinant. 

Properties of Determinants

There are various properties associated with determinants. These are used to simplify evaluating the determinant to find its value. Some of them are described below –

1. Expansion of determinants through any row or column gives the same value of the determinant. 

2. If a determinant A has a row or column where all elements are 0, then the value of the determinant will be 0. This is known as the All-Zero property.

3. If there exists a determinant with 2 identical rows and columns (same elements), then the determinant’s value is 0.

4. If a determinant has 2 rows or columns that are proportional, then the determinant value is 0. This is known as the Proportionality property.

5. If a determinant is transposed, then its value remains unchanged. For a determinant A, |A| = |A’| This is known as the reflection property. 

6. If 2 of the rows or columns are flipped, then the determinant value turns negative of the original value. This is known as the switching property. 

7. The determinant of an identity matrix is 1.

8. If a determinant A is multiplied by any constant value, then the value of the matrix thus obtained will be multiplied by the sa. This is called the Scalar Multiple Property. 

9. The product of the value of 2 determinants is equivalent to the product of two determinants. That is, det A det B = det (AB)

Conclusion

Determinant Matrices are an essential part of linear algebra. Matrices are an arrangement of numbers in rows and columns enclosed within a bracket. On the other hand, determinants are square matrices with a definite value. The overall subject matter adds to the calculation of adjoint and inverse of a matrix. This is used in solutions of linear equations and fields of physics, computer science, etc.