Properties of Matrix Multiplication

We have already discussed types of matrices on this blog. When we multiply matrices we should know about some of the basic properties of matrix multiplication. When you multiply matrices, you can cross check the multiplication online. In this article, we will take you through various properties of matrix multiplication. First, we will understand some basic terms associated with matrices:

Multiplication of a matrix by a scalar: Let us consider a matrix of order 3×3 and “K” be any number, then the product of the given matrix by a number “K” called scalar will be as shown below;

properties of matrix multiplication

Properties of matrix multiplication
  • Commutative law: Matrix multiplication is not commutative. Let A & B be the two matrices of some order, then we can say that;

AB ≠ BA

  • Distributive law: matrix multiplication is distributive. Let A, B & C be the three matrices of some order, then we can say that;

A(B+C) = AB + AC

also, (A+B)C = AC +BC

  • Associative law: Matrix multiplication is associative. Let A, B & C be the three matrices of some order, then we can say that;

(AB)C = A(BC)

  • The product of a matrix with a null matrix is always a null matrix. Let A is a matrix of order m×n and O is a null matrix of order n×p, then we have:

AO = O

  • If the product of any two matrices A & B is a null matrix, it is not necessary that one of the two matrices is a null matrix.
  • The product of any matrix with the identity matrix is the matrix itself.
  • If AB = O, it does not necessarily mean that BA = O, where A and B are any two matrices and O is a null matrix.

These are the basic properties of matrix multiplication. If you have any questions you can ask us in comments below or you can also contact us in the contact section.