How this instrument works
Matrix addition works entry by entry: the sum's row-1,column-1 cell is simply A's row-1,column-1 cell plus B's — same position, nothing more elaborate. That simplicity is not an accident; a matrix is fundamentally a rectangular array of numbers, and adding two arrays the obvious way, pairing each slot with the slot directly across from it, is the definition mathematicians settled on precisely because it makes the set of all matrices behave like ordinary vectors, obeying the same commutative and associative rules as adding coordinates in a plane.
Because the rule is purely positional, both matrices must share the exact same shape: a 2×2 only ever adds against another 2×2, since every cell needs a partner sitting in the identical spot, and a 2×3 simply has no row-2,column-3 entry to offer. Subtraction rides along for free — entering a negative value in any B field, say −5 instead of 5, turns that addition into a subtraction at that one cell, so this same instrument covers A − B without a separate mode or a second formula to learn.
The zero matrix is the quiet edge case worth knowing: adding it to any matrix changes nothing, the exact matrix equivalent of adding 0 to a number, and adding a matrix to its own negation collapses every cell to zero. Do not mistake this position-matched rule for how matrices multiply — matrix multiplication combines a whole row of A against a whole column of B and sums the products, demanding only that the inner dimensions agree rather than that the shapes match entirely, and it produces an entirely different kind of answer than this cell-by-cell sum.
- Enter matrix A's four entries into A: row1,col1, A: row1,col2, A: row2,col1, and A: row2,col2.
- Enter matrix B's four entries the same way, into B: row1,col1 through B: row2,col2 — to subtract instead of add, type B's values as negatives.
- Read the four Sum fields: Sum: row1,col1, Sum: row1,col2, Sum: row2,col1, and Sum: row2,col2 update the moment any entry changes.
- Change one A or B cell and confirm only the matching Sum cell moves — a quick proof that this addition never crosses rows or columns.
Worked example — [[1,2],[3,4]] plus [[5,6],[7,8]]
Set A: row1,col1 = 1, A: row1,col2 = 2, A: row2,col1 = 3, A: row2,col2 = 4, and B: row1,col1 = 5, B: row1,col2 = 6, B: row2,col1 = 7, B: row2,col2 = 8. Each Sum cell adds only its own matching pair: Sum: row1,col1 = 1 + 5 = 6, Sum: row1,col2 = 2 + 6 = 8, Sum: row2,col1 = 3 + 7 = 10, and Sum: row2,col2 = 4 + 8 = 12.
The full result reads S = [[6,8],[10,12]]. Multiply the same two starting matrices instead of adding them, using ordinary row-times-column matrix multiplication, and the answer is [[19,22],[43,50]] — nothing like this sum, because multiplication combines a whole row against a whole column rather than pairing matching positions. Addition's plainness is exactly what makes it the operation worth checking by hand before trusting anything more elaborate.
Questions
What is the formula for adding two matrices?
Add each entry of A to the entry sitting in the identical position of B: S₁₁ = A₁₁ + B₁₁, and so on for every cell. There is no summing across rows or columns, unlike matrix multiplication — for A = [[1,2],[3,4]] and B = [[5,6],[7,8]] that gives S = [[6,8],[10,12]], each cell an independent sum of just two numbers.
Can this calculator subtract matrices instead of adding them?
Yes — type any B entry as a negative number and that cell effectively subtracts. Entering B: row1,col1 as −5 instead of 5 turns S₁₁ = A₁₁ + B₁₁ into A₁₁ − 5, so A − B is really just A + (−B): the same addition rule applied to a negated matrix rather than a separate operation.
Why must both matrices be the same size to add them?
Every cell in the sum needs a partner occupying the identical row and column in the other matrix, so a 2×2 can only be added to another 2×2. A 2×3 has a row-2,column-3 entry that a 2×2 simply does not possess, leaving nothing for it to pair with — the operation is undefined the moment the shapes disagree, not merely inconvenient.
How is matrix addition different from matrix multiplication?
Addition and subtraction only ever touch matching positions, so S₁₁ depends purely on A₁₁ and B₁₁. Matrix multiplication instead sums a whole row of A against a whole column of B for each output cell — multiplying [[1,2],[3,4]] by [[5,6],[7,8]] gives [[19,22],[43,50]], nothing like the [[6,8],[10,12]] this sheet returns for the same two matrices added.
What happens when you add a matrix to the zero matrix?
Nothing changes — the zero matrix, every entry 0, is the additive identity, exactly the way adding 0 leaves an ordinary number untouched. Add a matrix to its own negation instead and every cell cancels to 0, since A + (−A) = 0 for the same reason 5 + (−5) = 0.
Does matrix addition follow the same rules as adding ordinary numbers?
Yes — it is commutative (A + B = B + A) and associative ((A + B) + C = A + (B + C)), because each cell is just two real numbers being added and ordinary addition already obeys both rules. This is also why matrices form what mathematicians call a vector space under addition: the same algebra governing plain numbers carries over entry by entry.