Addition and scaling are a short entrywise warm-up. The product is the new operation: the entry in row i and column j of AB is built from the whole of row i and the whole of column j, and the order of the factors changes the answer.
Core ideas
Two of the three operations behave exactly as the notation suggests. The third does not, and most of this lesson is about the third.
A + B exists only when A and B have the same number of rows and the same number of columns, and then it is what it looks like. Every rule of addition from Foundations of Algebra carries over, one position at a time.
Multiply each entry of row i of A by the matching entry of column j of B and add the results. The row length of A must equal the column height of B, and that requirement is the whole of the shape rule.
They can have different sizes; one can exist while the other does not; and even when both are n × n they are usually different matrices. Every rearrangement that quietly used commutativity has to be re-earned here.
The material
Addition and scaling in brief, then the shape rule, row-column products and the order of the factors.
An m × n matrix has m rows and n columns. Two matrices are equal when they have the same size and agree in every position. There is no notion of two matrices of different sizes being equal.
For matrices of the same size, A + B is the matrix whose entry in each position is the sum of the entries of A and B in that position. For a number k, the matrix kA has every entry of A multiplied by k.
A number acting on a matrix this way is called a scalar, to distinguish it from a 1 × 1 matrix, which is a different object.
Addition inherits everything from the numbers, position by position: A + B = B + A, (A + B) + C = A + (B + C), the zero matrix of the right size adds nothing, and A + (−A) is that zero matrix. Nothing here requires attention.
Let A be m × n and B be n × p. The product AB is the m × p matrix whose entry in row i, column j is a₁b₁ + a₂b₂ + … + aₙbₙ, where a₁, …, aₙ is row i of A and b₁, …, bₙ is column j of B.
If the row length of A is not the column height of B, the product does not exist. It is not zero and it is not approximately anything: there is no such matrix.
The shape rule is easier to use than to state. Write the two sizes side by side, (m × n)(n × p). If the inner pair match, they cancel and the outer pair give the size of the answer. If they do not match, stop.
The definition looks arbitrary until a system is written with it. Take the coefficient matrix of the system in “Systems in Three Variables”, a column of the three unknowns, and a column of the three constants. The product of the first two is a column whose entries are the left-hand sides of the three equations, so the whole system is the single statement Ax = b.
Even when both products exist, order must be preserved: AB and BA can have different sizes or different entries. The arithmetic lesson here is deliberately narrow — decide existence, state the output size, and build each entry from the named row and column. “Inverse Matrices” introduces the identity and inverse when a system needs a matrix operation to be undone.
Interactive
Sum, difference, scalar multiple and product, with every entry derived
Method
Four moves. The first one is written down, not done in your head.
(m × n)(n × p) gives m × p. If the inner numbers differ there is nothing to compute, and "the product does not exist" is the complete answer.
For the entry in row i, column j, run along row i of the left matrix and down column j of the right one, multiplying in pairs. Say the position out loud before starting it.
The number of products equals the shared inner dimension, so an inner dimension of 3 means three products every time. Counting them is a free check on whether a term was skipped.
If the question asks for BA, compute BA. Swapping the factors is not a simplification; it is a different calculation with a different answer.
Both matrices are 2 × 2, so both products exist and both are 2 × 2. They are still not the same matrix.
B here is a swapper. Multiplying A on the right by it exchanged the columns of A; multiplying on the left exchanged the rows. That is a fair picture of the general situation — which side you multiply on decides whether rows or columns are acted on — and it is why the order is part of the expression rather than a detail of how it was written down.
Neither product is the entrywise one, which would have been 0, 2, 3, 0. That matrix is not AB, is not BA, and answers no question asked in this course.
Faded rehearsal: let C be 2 × 3 with rows (1, 0, −2) and (3, 1, 4), and let D be 3 × 1 with column (2, −1, 5). The shape decision is supplied: CD exists and is 2 × 1. Compute its two entries by naming each row of C against the column of D, then explain from the dimensions why DC does not exist.
Before computing a single entry, write m × p down. It catches an impossible product before any arithmetic happens, it tells you how many entries you owe, and it is the habit that makes Ax = b in “Inverse Matrices” read as one equation rather than as three.
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