Zero Matrix

A zero matrix, also called a null matrix, is a matrix in which every element is zero. It can have any number of rows and columns, so it may be a square or rectangular matrix.

Some common examples of zero matrices of 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 the 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]

Solved Examples

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.