Zero matrix

As the name suggests, a ‘zero’ matrix is just a matrix where all the elements are zero.

Order of a zero matrix

We can have zero matrices of any order (dimensions), for example:

• A 2 \times 3 zero matrix:
  • \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}
• A 3 \times 2 zero matrix:
  • \begin{bmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{bmatrix}
• A 4 \times 4 zero matrix:
  • \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}
• A 1 \times 5 zero matrix:
  • \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \end{bmatrix}
• A 5 \times 1 zero matrix:
  • \begin{bmatrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}
• A 2 \times 2 zero matrix:
  • \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}

Identity matrix

A zero matrix is the identity matrix for addition. That’s because, if we add a zero matrix to any other matrix of the same order, we get that same matrix back.

For example, if we have the matrix \mathbf{A} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} and the zero matrix \mathbf{0} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, then:

• \mathbf{A} + \mathbf{0} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}
• \mathbf{0} + \mathbf{A} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} + \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

If you want some algebraic proof:

• Let \mathbf{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} and \mathbf{0} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, then:
• \mathbf{A} + \mathbf{0} = \begin{bmatrix} a + 0 & b + 0 \\ c + 0 & d + 0 \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \mathbf{A}
• \mathbf{0} + \mathbf{A} = \begin{bmatrix} 0 + a & 0 + b \\ 0 + c & 0 + d \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \mathbf{A}

Multiplication with a zero matrix

If we multiply any matrix by a zero matrix of compatible dimensions, we get a zero matrix as the result.

The proof of this for some 2 \times 2 matrices is as follows:

• Let \mathbf{A} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} and \mathbf{0} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}, then:
• \mathbf{A} \times \mathbf{0} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \times \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} a\times 0 + b\times 0 & a\times 0 + b\times 0 \\ c\times 0 + d\times 0 & c\times 0 + d\times 0 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \mathbf{0}
• \mathbf{0} \times \mathbf{A} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} \times \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} 0\times a + 0\times c & 0\times b + 0\times d \\ 0\times a + 0\times c & 0\times b + 0\times d \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \mathbf{0}