Linear Algebra 17 | Properties of the Matrix Product [dark version]

Matrix multiplication is built from a simple rule: each entry of the product comes from an inner product between a row of the first matrix and a column of the second. If A is an m×n matrix and B is an n×p matrix, then AB is m×p, and the (i,j) entry is computed by summing over the “middle” index: (AB)_{ij} = Σ_{ℓ=1..n} a_{iℓ} b_{ℓj}. This “sum over the middle” is the mechanism behind every algebraic property that follows, because it reduces matrix identities to familiar arithmetic with real numbers.

From that definition, several calculation rules carry over from ordinary arithmetic—especially distributive, scalar compatibility, and associativity. For distributivity, matrix multiplication distributes over addition when the addition happens on the same side: (A + B)C = AC + BC and A(B + C) = AB + AC, with the order of factors preserved. Scalar multiplication behaves consistently too. For any real scalar λ, multiplying a matrix product by λ from the left matches multiplying one factor by λ first: λ(AB) = (λA)B = A(λB). Associativity also holds: (AB)C = A(BC), provided the matrix sizes match so each product is defined.

The key reason these laws work is that the product’s definition turns each matrix entry into sums and products of real numbers. That lets standard real-number identities—like distributive rearrangements and regrouping of sums—transfer directly to the matrix setting. A component-wise proof of associativity tracks a fixed (i,j) entry on both sides, expands both products using the summation formula, and then rearranges sums and factors until the same expression appears for A(BC) and (AB)C.

One major arithmetic property does *not* carry over: commutativity. In general, AB ≠ BA because swapping the order changes which rows and columns get paired in the inner-product computation. A concrete 2×2 example makes the point: multiplying the matrix [[0,1],[-1,0]] by [[1,1],[1,1]] yields a different result than reversing the order. Associativity remains intact—parentheses can move without changing the outcome—but the order of the matrices themselves cannot be exchanged in general. That distinction—associative but not commutative—is the practical takeaway for doing matrix calculations correctly.