Zero Or Null Matrix

Matrices are one of the important concepts of algebra that also have applications in geometry. In geometry, matrices represent a geometric object or properties of a geometric object. Hence, a matrix is an arrangement of numbers or symbols in a particular fashion. There are various matrices based on various properties like shape and size. The type of elements can determine the type of matrix. A matrix whose all elements are zero is called a null matrix or a zero matrix.

Types of matrices based on elements present

The types of matrices based on the shape of a matrix are a square matrix, rectangular matrix and a triangular matrix. However, mattresses can also be classified based on what elements they contain. Following are the types of mattresses based on the elements they contain.

Unit matrix:

All the matrix elements are zero in the unit matrix, except one element, whose value is 1. Hence, it is called a unit matrix.

• All-ones matrix-

An all-ones matrix is a type of matrix where every element in the matrix is equal to 1.

• Identity matrix:

An identity matrix is a matrix in which all the diagonal elements are equal to 1, and all the other elements are zeros.

• Zero matrices:

A zero matrix is a type of matrix in which all the entries are zero. Let us learn in-depth about the zero matrices.

The zero matrix

The zero matrix is also known as the null matrix. It is named so because all the entries in a zero matrix are of the number zero. A zero matrix cannot contain elements like numbers or symbols other than zero.

If a is added to any other matrix B, the result will be the same matrix B. Hence, a null matrix is called the additive identity of a particular matrix. A null Matrix can have any number of rows and any number of columns. An example of a zero matrix is:

The above matrix is of the order 2×3 and contains 6 elements. All these elements have the value zero because it is a zero matrix.

Let’s take another zero matrix example.

Here the above matrix contains 1 row and 3 columns. There are 6 elements in total in the above Matrix, and all of them are 0. The above matrix is also a row matrix.

If the determinant of matrix A is zero, then the matrix is a zero matrix. 

The zero matrices are often confused with the zero diagonal matrices. In the zero diagonal matrices, all the diagonal elements of a matrix are equal to zero. The rest of the elements in the matrix could be any numbers. However, in a zero matrix or a null matrix, the rest of the matrix elements, along with the diagonal elements, must be zero.

Let us study the properties of a zero matrix.

 Properties of a zero matrix

The properties of a zero matrix are given below.

• The number of rows in a null matrix need not be equal to the number of columns in that matrix.

• The zero matrices are also singular since the determinant of the zero matrices is equal to zero.

• If a null matrix is added to any other Matrix A, the result will be the same matrix A, and no changes will be made.

• If any matrix A is multiplied with a null matrix, the result is another null matrix.

• As discussed above, the determinant of the zero matrices is equal to zero.

Set matrix zeroes

The set matrix zeroes are a type of problem asked about the matrices. Suppose any row or column contains the element zero; you are supposed to change all the elements in that row and the column to zero.

There are various approaches used to solve this problem. 

One of the methods used to solve the set matrix zeroes problems is

• The brute force approach:

In this method, an auxiliary matrix of the same size is created, and all its elements are equated to 1. Changes are made in the auxiliary Matrix and letter copy to the original matrix. After all zeroes in the original matrix are looked at,  the columns and rows in the auxiliary matrix are updated.

Conclusion:

Based on the elements that they contain, matrices can be classified into various types. Matrices can be classified as identity matrices, matrix of ones, zero matrices, etc. The zero matrix is also called the null matrix. All the entries in a zero matrix are required to be zero, not any other number of symbols. A zero matrix can have any number of rows and columns. If the determinant of matrix A is zero then that matrix could be a zero matrix. When zero matrices are added with any other matrix, the result is the same matrix.