How this instrument works
Matrix multiplication builds every entry of the product by pairing an entire row of the first matrix against an entire column of the second and summing those paired products — P₁₁ blends all of row 1 of A with all of column 1 of B, not merely two numbers sharing one position. Take A = [[1,2],[3,4]] and B = [[5,6],[7,8]]: row 1 of A is (1,2) and column 1 of B is (5,7), so P₁₁ = 1×5 + 2×7 = 19. Run the same two matrices through the Hadamard product instead — the operation that multiplies only matching positions — and the top-left entry comes out 1×5 = 5. Same starting matrices, two named operations, two different results, because one sums across a whole row-and-column pairing and the other never leaves a single cell.
The reason multiplication is built this way, and not by the simpler position-matching rule, is that a matrix stands for a rule turning one vector into another: Ax stretches, rotates, or shears x into a new vector. The product AB is defined so that (AB)x always equals A(Bx) for every vector x — apply B's transformation first, then feed the result through A. Composing two linear transformations has to produce another linear transformation, and the row-times-column recipe is exactly the bookkeeping that keeps that promise for every input at once, not just the one you happen to test by hand.
Order matters here in a way ordinary number multiplication never prepares anyone for: AB and BA generally land on different matrices, and for non-square shapes they may not even be the same size. Multiplication itself only exists when the first matrix's column count equals the second's row count — a rule two 2×2 matrices satisfy automatically, which is why this fixed shape hides a constraint that trips up the first encounter with rectangular pairs. Arthur Cayley formalized the operation in 1858 precisely so that multiplying matrices would mirror substituting one linear system into another, wiring composition into the algebra from the outset.
- Enter matrix A's four numbers into A: row1,col1, A: row1,col2, A: row2,col1, and A: row2,col2.
- Enter matrix B the same way, into B: row1,col1, B: row1,col2, B: row2,col1, and B: row2,col2.
- Read the four Product fields — Product: row1,col1 through Product: row2,col2 — each one a row of A dotted with a column of B.
- Swap A's and B's values and recompute to see AB and BA differ, a quick check that the order genuinely changes the answer.
Worked example — composing [[1,2],[3,4]] with [[5,6],[7,8]]
Set A to [[1,2],[3,4]] and B to [[5,6],[7,8]] — the same eight numbers as A: row1,col1 = 1 through B: row2,col2 = 8. Product: row1,col1 pairs A's top line against B's first column: 1×5 + 2×7 = 5 + 14 = 19. Product: row1,col2 pairs that same top line against B's second column: 1×6 + 2×8 = 6 + 16 = 22.
The bottom line of A follows the identical pattern: Product: row2,col1 = 3×5 + 4×7 = 15 + 28 = 43, and Product: row2,col2 = 3×6 + 4×8 = 18 + 32 = 50. The full product reads P = [[19,22],[43,50]] — a genuinely different table from the [[5,12],[21,32]] the same two matrices would give under the Hadamard, element-by-element rule, because every cell here blends four numbers instead of one.
Questions
How is matrix multiplication different from multiplying matrices entry by entry?
Matrix multiplication sums a whole row of A against a whole column of B for every output cell, so P₁₁ = A₁₁B₁₁ + A₁₂B₂₁ blends four numbers. The entry-by-entry rule — the Hadamard product — only multiplies the two values sharing one position, giving P₁₁ = A₁₁ × B₁₁ and nothing else. For A = [[1,2],[3,4]] and B = [[5,6],[7,8]] the two rules disagree everywhere: [[19,22],[43,50]] against [[5,12],[21,32]].
Why does the order of multiplication matter for matrices?
Because every output entry mixes a row of the first matrix with a column of the second, swapping which matrix goes first changes every sum being formed — AB and BA are built from different row-column pairings and generally land on different numbers, or aren't even defined for the same shapes. Ordinary numbers commute because there is only one way to pair two scalars; matrices offer many rows and columns to pair, and the recipe fixes one specific pairing per position.
What sizes of matrices can be multiplied together?
Two matrices can be multiplied only when the first matrix's column count equals the second's row count — an m×n matrix times an n×p matrix produces an m×p result. Both matrices here are fixed at 2×2, so that inner-dimension rule is satisfied automatically and never blocks a calculation, but it becomes the first thing to check once matrices stop being square.
What does matrix multiplication actually represent?
It represents composing two linear transformations — applying one, then the other, as a single combined transformation. If Bx maps a vector through B first and A(Bx) sends that result through A, the product matrix AB is built so that (AB)x gives the same two-step result directly. That is why the row-times-column recipe exists rather than a simpler position-matching rule: only the row-column form keeps the composition true for every input vector.
What mistake do people most often make with matrix multiplication?
Multiplying matrices position by position, the way you would multiply two ordinary number grids — that operation exists and is called the Hadamard product, but it is not standard matrix multiplication and gives a different answer for the same inputs. The second common slip is assuming AB equals BA; check both orders separately, since reversing them is a different calculation, not a rearrangement of the same one.
Does matrix multiplication have an identity element?
Yes — the identity matrix, with 1's on the diagonal and 0's elsewhere, leaves the other matrix unchanged: AI = A and IA = A. For this 2×2 sheet that means A: row1,col1 = 1, A: row1,col2 = 0, A: row2,col1 = 0, A: row2,col2 = 1 typed into either slot. Multiplication is also associative, so (AB)C always equals A(BC), even though AB and BA themselves generally differ.