Matrix Multiplication and its Different Properties

Matrices are one of the most powerful tools originating from the world of mathematics that can be used to solve many practical life problems. In this article, we will discuss how matrix multiplication is carried out and various other important things related to it. But before that, it is important that we discuss the concepts of matrices too. By mathematical definition, a matrix is an ordered rectangular arrangement of numbers or functions. The numbers or the functions present in the matrix are called the elements or the entries of the matrix. There are different types of matrices namely, square matrix, column matrix, row matrix, scalar matrix, identical matrix, and a lot more. Algebraic operations like matrix multiplication, addition, and subtraction are conducted on these matrices.

Matrices’ Order

A matrix M with the x rows and y columns is called a matrix of order (x,y) and is read as x by y. The order of a matrix plays an important part in determining various things.

Matrix Multiplication

The multiplication of XY of the two matrices X and Y can be defined only if the number of columns of Matrix X is equivalent to the number of rows in Matrix Y.

Properties of Matrix Multiplication

  • It is defined only when the number of columns in the first matrix is equal to the number of rows in the 2nd matrix.
  • It is not commutative in general, i.e. AB is not equal to BA
  • It is associative in nature, i.e. (AB) C = A (BC)
  • Matrix multiplication by scalar k, implies the multiplication of each and every element.
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Different Categories of Matrices

  • Row Matrices: Row matrices are those which consist of a single row. It is also called a row vector.
  • Column Matrices: Column matrices are those which consist of a single column. It is also called a column vector.
  • Null Matrix or Zero Matrix: A matrix whose each and every element is zero is called a null matrix or a zero matrix.
  • Square Matrix: A square matrix is a matrix whose numbers of the rows are the same as that of the number of columns. It is a very special type of matrix since the concept of determinants uses a square matrix only.
  • Rectangular Matrix: A rectangular matrix is a matrix whose numbers of rows are different from that of the number of columns.
  • Scalar Matrices: The matrices whose top diagonal elements are all same or equal are called scalar matrices.
  • Unit Matrices: Unit matrices are a unique type of scalar matrices whose diagonals are equal to one.
  • Upper Triangular Matrices: The matrices in which all the elements below the leading diagonal are zero are called the upper triangular matrices.
  • Lower Triangular Matrices: The matrices in which all the elements above the leading diagonal are zero are called the lower triangular matrices.
  • Sub Matrix: A submatrix is a matrix that is obtained by deleting or removing one or more columns or rows or both of a matrix. 
  • Equal Matrices: Two matrices are said to be equal if they satisfy the following conditions: the order of both the matrices is the same and the corresponding elements in both the matrices are equal.

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