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Matrix Addition — Definition, Formula & Examples

Matrix Addition

Matrices are added term-by-term as shown below.

Key Formula

If A=[aij] and B=[bij], then A+B=[aij+bij]\text{If } A = [a_{ij}] \text{ and } B = [b_{ij}], \text{ then } A + B = [a_{ij} + b_{ij}]
Where:
  • aija_{ij} = The entry in row i, column j of matrix A
  • bijb_{ij} = The entry in row i, column j of matrix B

Worked Example

Problem: Find A + B given A = [[1, 3], [5, 7]] and B = [[2, 4], [6, 8]].
Step 1: Confirm both matrices have the same dimensions. A is 2×2 and B is 2×2, so addition is defined.
Step 2: Add each corresponding entry: top-left with top-left, top-right with top-right, and so on.
A+B=[1+23+45+67+8]A + B = \begin{bmatrix} 1+2 & 3+4 \\ 5+6 & 7+8 \end{bmatrix}
Step 3: Compute each sum to get the resulting matrix.
A+B=[371115]A + B = \begin{bmatrix} 3 & 7 \\ 11 & 15 \end{bmatrix}
Answer: A + B = [[3, 7], [11, 15]]

Why It Matters

Matrix addition is foundational in linear algebra, computer graphics, and data science. When you represent data as matrices—such as pixel values in an image or coordinates in a transformation—adding matrices lets you combine or overlay that information. It also underpins more advanced operations like solving systems of linear equations.

Common Mistakes

Mistake: Trying to add matrices of different dimensions, such as a 2×3 matrix and a 3×2 matrix.
Correction: Matrix addition is only defined when both matrices have the exact same number of rows and the exact same number of columns. If the dimensions don't match, the sum does not exist.

Related Terms