Zero or null matrix

On this post you will find the explanation of the zero matrix, also known as null matrix. You will also see why the values of its entries are independent of the dimension of the matrix, and finally you will find all the properties of this type of matrix.

What is the zero (or null) matrix?

The meaning of zero (or null) matrix is as follows:

In mathematics, the zero matrix, also called null matrix, is a matrix which all its elements are equal to zero. The zero matrix is denoted by the symbol O or 0.

Therefore, according to the definition of the zero (or null) matrix, this matrix can be of all possible dimensions as long as all its numbers are zero. Take a look at the following examples:

Examples of null matrices

Example of a 2×2 zero matrix

example of 2x2 null or zero matrix

Example of a 3×3 zero matrix

example of 3x3 null or zero matrix

Example of a 4×4 zero matrix

example of 4x4 null or zero matrix

The above examples are all square matrices, but zero matrices can also be rectangular.

Perhaps now it seems to you that this peculiar matrix is not important, since it is simply a matrix full of zeros. However, in mathematics, particularly in the field of linear algebra, it is a very useful matrix because it makes calculations much easier.

➤ See also: lower triangular matrix

Properties of the zero or null matrix

Zero or null matrices have the following characteristics:

  • The zero matrix is the additive identity of the sum matrix operation, therefore:

 \displaystyle A + 0 = A

  • Matrix multiplication has the multiplicative property of zero, that is, the product of any matrix multiplied by the null matrix is equal to 0.

 \displaystyle A \cdot 0 = 0 \cdot A = 0

  • The null matrix is the only matrix whose rank is zero.
  • The determinant of the null (or zero) matrix always results in 0, so this type of matrix is a singular matrix.

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