Zero Matrix

A zero matrix (also known as a null matrix) is a matrix in which all elements are zero. It is denoted by \( O \) and can be of any size \( m \times n \).

Mathematically, a zero matrix of order \( m \times n \) is represented as:

\[ O_{m \times n} = \begin{bmatrix} 0 & 0 & \dots & 0 \\ 0 & 0 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 0 \end{bmatrix} \]

For example:

  • A \( 2 \times 2 \) zero matrix: \[ O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]
  • A \( 3 \times 3 \) zero matrix: \[ O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]

Properties of Zero Matrix

1. Additive Identity Property

The zero matrix acts as the additive identity in matrix addition. Adding a zero matrix to any matrix \( A \) of the same order does not change \( A \).

\[ A + O = O + A = A \]

Example:

Given:

\[ A = \begin{bmatrix} 5 & -3 \\ 2 & 7 \end{bmatrix}, \quad O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

\[ A + O = \begin{bmatrix} 5 & -3 \\ 2 & 7 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 5 & -3 \\ 2 & 7 \end{bmatrix} \]

2. Multiplication by a Scalar

Multiplying the zero matrix by any scalar \( k \) results in another zero matrix:

\[ kO = O \]

Example:

Let:

\[ O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, \quad k = 3 \]

\[ 3O = 3 \times \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]

3. Multiplicative Property

Multiplying any matrix \( A \) by a zero matrix of appropriate dimensions results in a zero matrix.

  • If \( A \) is an \( m \times n \) matrix and \( O \) is an \( n \times p \) zero matrix, then: \[ A \cdot O = O_{m \times p} \]
  • Similarly, if \( O \) is an \( m \times n \) zero matrix, and \( B \) is an \( n \times p \) matrix, then: \[ O \cdot B = O_{m \times p} \]

Example:

Let:

\[ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

\[ A \cdot O = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \cdot \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

4. Determinant of a Zero Matrix

The determinant of a square zero matrix of order \( n \) is always zero.

\[ \det(O) = 0 \]

Example:

For a \( 2 \times 2 \) zero matrix:

\[ O = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \]

The determinant is:

\[ \det(O) = (0 \times 0) – (0 \times 0) = 0 \]

5. Zero Matrix and Inverse

The zero matrix has no inverse because its determinant is always zero, making it a singular matrix.

\[ O^{-1} \text{ does not exist} \]

Example:

For the \( 3 \times 3 \) zero matrix:

\[ O = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \]

Since \( \det(O) = 0 \), the matrix is non-invertible.

6. Zero Matrix in Linear Equations

If a system of linear equations has a coefficient matrix that is a zero matrix, then the equations reduce to:

\[ 0x + 0y + 0z = 0 \]

which always holds true, but does not provide any useful information about the variables.

Example:

Consider the system:

\[ \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]

This results in trivial solutions but does not provide unique values for \( x \) and \( y \).