ZERO MATRIX

A zero matrix, also known as a null matrix, is a matrix with all of its entries equal to zero in mathematics, particularly linear algebra. It also acts as the additive identity of the additive group of m x n matrices and is indicated by the sign O or, depending on the context, subscripts corresponding to the matrix dimension.

The organisation of zero elements into rows and columns is known as a zero matrix. A zero matrix is one in which all of the entries are 0. It is represented by the letter ‘O,’ which can be interpreted as a subscript to reflect the matrix’s dimension.

What is a zero matrix?

A zero matrix is a type of matrix in which all of the elements are equal to zero. A zero matrix is also known as a null matrix because it has solely zeros as its elements. A square matrix can be made out of a zero matrix.

The letter ‘O’ stands for a zero matrix. When an additive identity matrix is added to a matrix of order m x n, the outcome is the same matrix.

0m×n =[ 0 ] 

Addition of zero matrix:-

When the non-zero matrix of order m x n  is added to a zero matrix of order m x n then the result will be the original matrix.

Let us suppose we have a matrix Am×n = [aij] is an

m × n matrix and 0 is a zero matrix of m× n order. Then.  A + 0 = 0 + A = A

When the zero matrix is added to another matrix, the identity of the matrix remains unchanged. As a result, it’s known as the additive identity for matrix addition.

Product of matrix with zero matrix:-

It is feasible to get a zero matrix by multiplying two non-zero matrices together.

If xy = 0, we can state that either x = 0 or y = 0 given two real numbers, say x and y. Matrixes can be thought of in the same way.

The product of the two matrices A and B will result in a zero matrix if the rows of matrix A have zero items and the columns of matrix B which is of the same order having zero elements.

Properties of zero matrix:-

Some of the null matrix’s most important features are listed below.

• A square matrix is used as the null matrix.

• The number of rows and columns in the null matrix can be uneven.

• The addition of a null matrix to any matrix has no effect on the matrix’s properties.

• When a null matrix is multiplied by another null matrix, the result is a null matrix.

• A null matrix’s determinant is equal to zero.

• A singular matrix is a null matrix.

• The only matrix with a rank of 0 is the zero matrix.

•  A – 0 = A

•  A – A = 0

•  0A = 0

• If cA = 0 then c = 0 or A = 0

Application of zero matrix :-

Simple solutions to algebraic equations involving matrices are possible with Zero Matrices. The zero matrix, for example, can be defined as an additive group, making it a useful variable in situations when an unknown matrix must be solved.

Consider the equation X + Y = Z, where X is the variable. To begin, simplify the equation to X + 0 = Z + (-Y). Y and -Y become a zero matrix when the inverse matrix is added to either side of the equation. The additive identity property thus reduces X + 0 to merely X, yielding X = Z – Y. Algebraic equations are substantially easier to calculate when the zero matrix is used.

CONCLUSION:-

A null matrix is a square matrix in which all of the elements are 0. Because the null matrix’s elements are all zeros, the null matrix is also known as a zero matrix. Any matrix’s additive identity is the null matrix. A null matrix has the order m x n and can have an uneven number of rows and columns.

The linear transformation that transfers all vectors to the zero vector is also represented by the zero matrix.  It is idempotent, which means that when multiplied by itself, the result is the same as the original. The only matrix with a rank of 0 is the zero matrix.