Matrix addition

We can only add together matrices with the same order.

Adding matrices

To add matrices, simply add up the corresponding elements from each matrix.

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

• We add the corresponding elements:
  • Top-left: 1 + 5 = 6
  • Top-right: 2 + 6 = 8
  • Bottom-left: 3 + 7 = 10
  • Bottom-right: 4 + 8 = 12
• So, \mathbf{A} + \mathbf{B} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}
• Answer: \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}

Find \mathbf{C} + \mathbf{D} where C=\begin{bmatrix} 2 & 4 & 6 \\ 8 & 10 & 12 \end{bmatrix} and D=\begin{bmatrix} 1 & 3 & 5 \\ 7 & 9 & 11 \end{bmatrix}

• We add the corresponding elements:
  • Top-left: 2 + 1 = 3
  • Top-middle: 4 + 3 = 7
  • Top-right: 6 + 5 = 11
  • Bottom-left: 8 + 7 = 15
  • Bottom-middle: 10 + 9 = 19
  • Bottom-right: 12 + 11 = 23
• So, \mathbf{C} + \mathbf{D} = \begin{bmatrix} 3 & 7 & 11 \\ 15 & 19 & 23 \end{bmatrix}
• Answer: \begin{bmatrix} 3 & 7 & 11 \\ 15 & 19 & 23 \end{bmatrix}

Commutative

• Matrix addition is commutative.
• We can add matrices in any order:
  • \mathbf{A} + \mathbf{B} = \mathbf{B} + \mathbf{A}

Associative

• Matrix addition is associative.
• We can group matrices in any way when adding:
  • (\mathbf{A} + \mathbf{B}) + \mathbf{C} = \mathbf{A} + (\mathbf{B} + \mathbf{C})