MatrixAdd

Matrix Addition Calculator

Enter two 2×2 matrices — see the sum with each element calculated step by step.

Matrix A (2x2)

Matrix B (2x2)

A + B

[[6, 8], [10, 12]]

What you entered

Matrix A: [[1, 2], [3, 4]]Matrix B: [[5, 6], [7, 8]]

1
C[0][0] = A[0][0] + B[0][0]
1 + 5= 6
2
C[0][1] = A[0][1] + B[0][1]
2 + 6= 8
3
C[1][0] = A[1][0] + B[1][0]
3 + 7= 10
4
C[1][1] = A[1][1] + B[1][1]
4 + 8= 12

Result

A + B: [[6, 8], [10, 12]]

Matrix addition adds corresponding elements: each entry in the result is the sum of the matching entries from A and B.

Adding 2x2 matrices

Matrix addition works entry by entry, so both matrices must have the same dimensions.

Matrix A	Matrix B	A + B
[[1,2],[3,4]]	[[5,6],[7,8]]	[[6,8],[10,12]]
[[2,0],[0,2]]	[[1,1],[1,1]]	[[3,1],[1,3]]
[[1,0],[0,1]]	[[3,4],[5,6]]	[[4,4],[5,7]]

Addition is the well-behaved matrix operation: it is commutative, so A + B equals B + A, and associative, exactly like ordinary addition. That is a marked contrast with multiplication, where order changes the answer. The requirement that dimensions match is strict - a 2x2 cannot be added to a 2x3 at all, whereas multiplication only requires the inner dimensions to agree. Subtraction works the same way, entry by entry.

Matrix addition rules

Matrix addition is done element by element: C[i][j] = A[i][j] + B[i][j]. Both matrices must have the same dimensions. Unlike matrix multiplication, addition is commutative (A + B = B + A) and straightforward.

Where matrix addition is used

Matrix addition appears in image processing (blending two images), physics (superposition of forces or fields), computer graphics (combining transformations), and statistics (adding covariance matrices).

Frequently asked questions

I'm blending two grayscale images represented as matrices — how does that work?

Each pixel is a number (0–255). To blend 50/50, add the two matrices element-by-element and divide by 2. For a weighted blend (70% image A, 30% image B), compute 0.7A + 0.3B element-wise. Matrix addition is the mathematical foundation of image compositing.

How is matrix addition different from matrix multiplication?

Addition is element-by-element (same position, just add). Multiplication uses dot products of rows and columns — C[i][j] involves an entire row of A and an entire column of B. Addition requires same-sized matrices; multiplication requires A's column count to match B's row count. Use the matrix multiplication calculator for that operation.

Can I add a 2×3 matrix to a 3×2 matrix?

No — matrix addition requires both matrices to have the exact same dimensions. A 2×3 and 3×2 matrix have different shapes. You can multiply them (2×3 times 3×2 gives 2×2), but not add them.

Is matrix addition commutative?

Yes — A + B = B + A, always. This is unlike matrix multiplication, where AB ≠ BA in general. Matrix addition is also associative: (A + B) + C = A + (B + C). These properties make addition the simpler of the two operations.