Matrix subtraction

We can only subtract matrices with the same order.

Subtracting matrices

To subtract matrices, simply subtract the corresponding elements from each matrix.

Find \mathbf{A} - \mathbf{B} where A=\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} and B=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

• We subtract the corresponding elements:
  • Top-left: 5 - 1 = 4
  • Top-right: 6 - 2 = 4
  • Bottom-left: 7 - 3 = 4
  • Bottom-right: 8 - 4 = 4
• So, \mathbf{A} - \mathbf{B} = \begin{bmatrix} 4 & 4 \\ 4 & 4 \end{bmatrix}
• Answer: \begin{bmatrix} 4 & 4 \\ 4 & 4 \end{bmatrix}

Find the value of \mathbf{C} - \mathbf{D} where C=\begin{bmatrix} 10 & 20 \\ 30 & 40 \end{bmatrix} and D=\begin{bmatrix} 5 & 15 \\ 25 & 35 \end{bmatrix}

• We subtract the corresponding elements:
  • Top-left: 10 - 5 = 5
  • Top-right: 20 - 15 = 5
  • Bottom-left: 30 - 25 = 5
  • Bottom-right: 40 - 35 = 5
• So, \mathbf{C} - \mathbf{D} = \begin{bmatrix} 5 & 5 \\ 5 & 5 \end{bmatrix}
• Answer: \begin{bmatrix} 5 & 5 \\ 5 & 5 \end{bmatrix}

Find the value of \mathbf{E} - \mathbf{F} - \mathbf{G} where E=\begin{bmatrix} 15 & 25 \\ 35 & 45 \end{bmatrix}, F=\begin{bmatrix} 5 & 10 \\ 15 & 20 \end{bmatrix} and G=\begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}

• First, we find \mathbf{E} - \mathbf{F}:
  • Top-left: 15 - 5 = 10
  • Top-right: 25 - 10 = 15
  • Bottom-left: 35 - 15 = 20
  • Bottom-right: 45 - 20 = 25
• So \mathbf{E} - \mathbf{F} = \begin{bmatrix} 10 & 15 \\ 20 & 25 \end{bmatrix}
• Then we do that value minus \mathbf{G}:
  • Top-left: 10 - 2 = 8
  • Top-right: 15 - 4 = 11
  • Bottom-left: 20 - 6 = 14
  • Bottom-right: 25 - 8 = 17
• So, \mathbf{E} - \mathbf{F} - \mathbf{G} = \begin{bmatrix} 8 & 11 \\ 14 & 17 \end{bmatrix}
• Answer: \begin{bmatrix} 8 & 11 \\ 14 & 17 \end{bmatrix}

Commutative

• Matrix subtraction is not commutative, which means that \mathbf{A} - \mathbf{B} \neq \mathbf{B} - \mathbf{A} (in general).
• FOr example, if A=\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} and B=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, then
  • \mathbf{A} - \mathbf{B} = \begin{bmatrix} 4 & 4 \\ 4 & 4 \end{bmatrix}
  • \mathbf{B} - \mathbf{A} = \begin{bmatrix} -4 & -4 \\ -4 & -4 \end{bmatrix}
• The results will always be the negative variant of each other.

Associative

• Matrix subtraction is also not associative, which means that \mathbf{A} - (\mathbf{B} - \mathbf{C}) \neq (\mathbf{A} - \mathbf{B}) - \mathbf{C} (in general).
• For example, if A=\begin{bmatrix} 15 & 25 \\ 35 & 45 \end{bmatrix}, B=\begin{bmatrix} 5 & 10 \\ 15 & 20 \end{bmatrix} and C=\begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}, then
  • \mathbf{A} - (\mathbf{B} - \mathbf{C}) = \begin{bmatrix} 12 & 19 \\ 28 & 37 \end{bmatrix}
  • (\mathbf{A} - \mathbf{B}) - \mathbf{C} = \begin{bmatrix} 8 & 11 \\ 14 & 17 \end{bmatrix}
• The results are different, so it’s not associative.

Distributive

• Matrix subtraction is distributive over matrix addition, which means that \mathbf{A} - (\mathbf{B} + \mathbf{C}) = (\mathbf{A} - \mathbf{B}) - \mathbf{C}.
• For example, if A=\begin{bmatrix} 15 & 25 \\ 35 & 45 \end{bmatrix}, B=\begin{bmatrix} 5 & 10 \\ 15 & 20 \end{bmatrix} and C=\begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}, then
  • \mathbf{A} - (\mathbf{B} + \mathbf{C}) = \begin{bmatrix} 8 & 11 \\ 14 & 17 \end{bmatrix}
  • (\mathbf{A} - \mathbf{B}) - \mathbf{C} = \begin{bmatrix} 8 & 11 \\ 14 & 17 \end{bmatrix}
• The results are the same, so it’s distributive.