Special Matrices

Matrix is the arrangement of numbers in the rows and columns. It is used to store and keep the numbers in the sequence of rows and columns. There are various types of matrices, which are different from each other in terms of their operations and functions. Although, there are some special cases in matrices, called special matrices. These special matrices have some distinctive properties, due to which they play an important role in various computations related to mathematics. Although, the famous mathematicians put five matrices under the category of special matrices, which we describe further in the article. 

Special matrices

The special matrices, which can be called special case matrices, have some distinctive properties. These matrices work as a special function in solving the problems and many more. Apart from this, these matrices are also used in various fields and subjects other than mathematics. In matrices there are five special types of matrices. 

Special types of matrices 

Let’s understand the special types of matrices in detail. 

• Square matrix 

• Identity matrix 

• Diagonal matrix 

• Symmetric matrix 

• Triangular matrix 

Square matrix 

In special square matrices, the number and count of columns remain equal to the number of columns. For instance, if the number of elements in the row is 5, then the number of elements in the column will also remain the 4 under this matrix. If the number of rows and columns is not equal, the matrix is not a square Matrix. 

Identity matrix 

The identity matrix is a special case matrix in which all the elements of the diagonals are one. Along with this, the elements other than the diagonal will remain zero. It is used to find the values of various matrix functions. 

Diagonal matrix 

The diagonal matrix is a type of special matrix. In a diagonal matrix, all the elements of the diagonal consist of a numerical value, while all other elements of the matrix remain zero. In this matrix, all the computations are done through the diagonal element.

Symmetric matrix 

These special matrices are defined as matrix elements whose elements form the mirror image of each other. In a symmetric matrix, all the angles are similar to the angles which fall in their front. 

Triangular matrix 

Triangular matrices are those special matrices that form a triangle’s shape. These matrices are of two types. These are:

• Lower triangular matrix 

• Upper triangular matrix 

Lower triangular matrix 

Lower triangular matrices are those matrices whose all elements are written below the diagonal. All other elements of this matrix remain zero. Although performing any type of computation on a lower triangular matrix, those elements are not considered.  

Upper triangular matrix 

Upper triangular matrices are those matrices whose all elements are written above the diagonal. All other elements of this matrix remain zero. 

Determinant of special matrices

Let’s understand the Determinant of special matrices. 

The value of the Determinant of special matrices is obtained using an easy method. One row or column is taken to find the special matrices’ value. After taking the one row and column, the first element of that particular row or column is taken. Then, all the elements related to the taken element in the row or column of that element get neglected.

 After that, all the elements are calculated further by performing cross multiplication. In this cross multiplication, the result is written separately. The same procedure will be repeated with all the selected row or column elements. After that, all the resulting values are computed. Some values are written with a positive sign, and some are written with a negative sign. The sign of these values is calculated by assigning the serial number of rows and columns and then putting the added value to the power of -1. Ii

If the values after adding the serial number are positive, the power of (-1) will also be positive, and the number will become positive. If the addicting comes odd, then the power of (-1) becomes negative. Due to this, while computing determinants, some values will remain positive and some negative. After performing basic operations, the value of the Determinant of special matrices will come.

Conclusion 

Matrix is the broad study of arranging the elements in the sequence of rows and columns. All the elements of the matrices are also called entries of the matrix. There are various types of matrix-like row matrix, column matrix, Skew -Symmetric Matrix, horizontal matrix, vertical matrix, etc. But, the Square matrix, Identity matrix, Diagonal matrix, Symmetric matrix and Triangular matrix are special types of matrices. These matrices consist of special characteristics. The properties of special matrices are slightly different from those of matrices. In simple words, the properties and computational methods of special matrices are modified and standard.