Zero Matrix

A zero matrix, or null matrix, is a matrix whose all elements are zeros. A matrix is defined as a rectangular array of numbers that are arranged in rows and columns. The size of a matrix can be determined by the number of rows and columns in it. A matrix is said to be an "m by n" matrix when it has "m" rows and "n" columns and is written as an "m × n" matrix. For example, the matrix given below is a "2 × 3" matrix, i.e., a matrix that has two rows and three columns. We have different types of matrices, such as rectangular matrices, square matrices, triangular matrices, symmetric matrices, etc.

A = \left[\begin{array}{ccc} 1 & -5 & 3\\ 7 & 8 & 4 \end{array}\right]

What is a Zero Matrix (Null Matrix)?

A zero matrix, or null matrix, is a matrix whose all elements are zeros. As a null matrix has all zeros as its elements, it is referred to as a zero matrix. A zero matrix can be a square matrix, or it can also have an unequal number of rows and columns.

A zero matrix is represented as "O." If we add a zero matrix to another matrix A of the same order, then the resultant matrix is A. So, a zero matrix is known as the additive identity of that particular matrix. The matrix given below represents a zero matrix of order "m by n."

O m×n = \left[\begin{array}{cccccc} 0 & 0 & . & . & . & 0\\ 0 & 0 & . & . & . & 0\\ . & . & . &  &  & .\\ . & . &  & . &  & .\\ 0 & 0 & . & . & . & 0 \end{array}\right]_{m\times n}

Examples of Zero Matrices

Some common examples of  zero matrices of the different orders are given below:

• Zero Matrix of order (1 × 1) → P_1,1  = [0]
• Zero Matrix of order (1 × 2) → P_1,2 = [0, 0]

Properties of a Zero Matrix

Important properties of a Zero Matrix are:

• A zero matrix can be a square matrix or a rectangular matrix, i.e., it can have an unequal number of rows and columns.
• As the determinant of a zero matrix is zero, a zero matrix is a singular matrix. (Also read about, How to Find Determinant of a Matrix?)
• If a zero matrix is added to another matrix A of the same order, then the resultant matrix is A.

A + O = O + A = A

• If a zero matrix is multiplied by another matrix A, then the resultant matrix is a zero matrix.

A × O = O × A = O

• If any matrix A is subtracted from itself, then the resultant matrix is a zero matrix.

A − A = O

• The determinant of a zero matrix or a null matrix is zero.

Addition of Zero Matrix

When a zero matrix of order "m by n" is added to another non-zero matrix A of the same matrix, then the resultant matrix is A.
Let A = [aij]_m×n be a non-zero matrix and O be a zero matrix of order "m by n," then

A + O = O + A = A

Example:

\left[\begin{array}{cc} 1 & 2\\ 3 & 4 \end{array}\right]+\left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right]=\left[\begin{array}{cc} 1 & 2\\ 3 & 4 \end{array}\right]

Article Realated to Zero Matrix:

• Row Matrix
• Column Matrix
• Rectangular Matrix
• Square Matrix
• Diagonal Matrix

Solved Examples on Zero Matrix

Example 1: Give an example of a zero matrix that has three rows and four columns.

Solution:

Order of a zero matrix that has three rows and four columns is "3 × 4" and all its elements are zero. The matrix given below represents a zero matrix of order "3 × 4."

O_3×4 = \left[\begin{array}{cccc} 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{array}\right]_{3\times4}

Example 2: Prove that if the product of two matrices is a zero matrix, then one of the matrices doesn't need to be a zero matrix.

Solution:

Let A = \left[\begin{array}{cc} 0 & 0\\ 2 & 0 \end{array}\right]   and B = \left[\begin{array}{cc} 0 & 0\\ 1 & 0 \end{array}\right]   be two non-zero matrices.

A × B = \left[\begin{array}{cc} 0 & 0\\ 2 & 0 \end{array}\right]\left[\begin{array}{cc} 0 & 0\\ 1 & 0 \end{array}\right]=\left[\begin{array}{cc} \left(0\times0+0\times1\right) & \left(0\times0+0\times0\right)\\ \left(2\times0+0\times1\right) & \left(2\times0+0\times0\right) \end{array}\right]

A × B = \left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right]   = O

Hence proved.

Example 3: Prove that a zero matrix is a singular matrix.

Solution:

To prove that a zero matrix is a singular matrix, let us consider a zero matrix of order "2 × 2."

O_2×2 = \left[\begin{array}{cc} 0 & 0\\ 0 & 0 \end{array}\right]

We know that,

The determinant of a matrix  \left[\begin{array}{cc} a & b\\ c & d \end{array}\right]  = ad - bc

So, the determinant of O_2×2 = 0 × 0 - 0 × 0 = 0 − 0 = 0

We know that a singular matrix is a matrix whose determinant is zero. As the determinant of a zero matrix is zero, a zero matrix is a singular matrix.

Hence proved.

Example 4: Prove that the additive identity of A = \left[\begin{array}{ccc} 1 & 5 & 9\\ 2 & 8 & 3 \end{array}\right]   is a zero matrix.

Solution:

To prove that, additive of the given matrix A is a zero matrix, we need to prove that

A + O = A

Given matrix A = 

\left[\begin{array}{ccc} 1 & 5 & 9\\ 2 & 8 & 3 \end{array}\right]+\left[\begin{array}{ccc} 0 & 0 & 0\\ 0 & 0 & 0 \end{array}\right]

=\left[\begin{array}{ccc} 1+0 & 5+0 & 9+0\\ 2+0 & 8+0 & 3+0 \end{array}\right]

= \left[\begin{array}{ccc} 1 & 5 & 9\\ 2 & 8 & 3 \end{array}\right] = A

Hence proved.