Use matrices to represent data and networks, and add, subtract, scalar-multiply and multiply matrices of appropriate dimensions

Learning goal: Use matrices to represent and manipulate data, such as payoffs in a game or incidence relationships in a network; multiply matrices by scalars to produce new matrices, as when all the payoffs in a game are doubled; and add, subtract and multiply matrices of appropriate dimensions, stating when each operation is defined.

Scope note. N.VM.6, N.VM.7 and N.VM.8 are marked (+) in Ohio's Learning Standards for mathematics, which Ohio uses to indicate standards "for students who are planning on taking advanced courses". They are taught here in full, as extension content for students on that path.

What a matrix is. A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with m rows and n columns has dimensions m × n, read "m by n", rows first. The entry in row i and column j is written a_ij. Matrices are named with capital letters, A, B, M. A 2 × 3 matrix has 6 entries; a 3 × 1 matrix is a single column, which is how a vector is written when it is treated as a matrix; a 1 × 1 matrix is just a number in brackets. The dimensions are part of the matrix's identity: a 2 × 3 and a 3 × 2 matrix are different kinds of object even if they hold the same six numbers.

Matrices as data tables. A matrix is a table with the row and column headings stripped off but remembered. Suppose a shop with two branches records how many of three products it sold in one week: branch 1 sold 12 notebooks, 8 pens and 5 folders; branch 2 sold 7, 10 and 9. That is the 2 × 3 sales matrix S with rows [12 8 5] and [7 10 9]. The rows mean branches and the columns mean products, and once that is fixed, every operation on S has a meaning in the shop.

Payoff matrices. In a simple two-player game, a payoff matrix lists what one player gains for each pair of choices: rows are that player's options and columns are the opponent's. A 2 × 2 payoff matrix with rows [3 −1] and [0 2] says: if I choose option 1 and my opponent chooses option 1, I gain 3; option 1 against option 2, I lose 1; and so on. The matrix is a compact record of every possible outcome.

Adjacency matrices for networks. The incidence relationships in a network (which node is directly linked to which) are recorded in an adjacency matrix, with one row and one column per node: entry (i, j) is 1 if there is a direct route from node i to node j and 0 if not. Take four towns with direct roads 1 → 2, 1 → 3, 2 → 3, 2 → 4 and 3 → 4. The matrix M has rows [0 1 1 0], [0 0 1 1], [0 0 0 1] and [0 0 0 0]. Row 4 is all zeros because no road leaves town 4; column 1 is all zeros because none arrives at town 1. The matrix says everything the map says.

Scalar multiplication. Multiplying a matrix by a scalar multiplies every entry by that scalar, producing a new matrix of the same dimensions. If all the payoffs in the game are doubled, the payoff matrix becomes 2 × [3 −1; 0 2] = [6 −2; 0 4]. If the shop expects sales to grow by 50% at both branches, the projected matrix is 1.5S, with rows [18 12 7.5] and [10.5 15 13.5]. The meaning of each entry is preserved; only its size changes.

Addition and subtraction. Two matrices can be added or subtracted only when they have the same dimensions, and then the operation is entry by entry: (A + B)_ij = a_ij + b_ij. If the next week's sales are T with rows [15 6 9] and [5 12 8], the fortnight's total is S + T with rows [27 14 14] and [12 22 17], and the change from week one to week two is T − S with rows [3 −2 4] and [−2 2 −1], negative entries meaning a fall. Adding a 2 × 3 matrix to a 3 × 2 matrix is undefined: there is no way to pair up the entries, and no shop meaning either.

Matrix multiplication: the dimension rule. The product AB is defined only when the number of columns of A equals the number of rows of B. If A is m × n and B is n × p, then AB is m × p: the inner dimensions must match and cancel, and the outer dimensions survive. The entry in row i, column j of AB is found by running along row i of A and down column j of B, multiplying the pairs and adding: (AB)_ij = a_i1 b_1j + a_i2 b_2j + … + a_in b_nj.

A fully worked model. Suppose the three products sell for $4, $6 and $10. Put the prices in a 3 × 1 column matrix P = [4; 6; 10]. Then SP is (2 × 3)(3 × 1), the inner 3s match, and the product is 2 × 1: one number per branch. Row 1 of S with column P: 12 × 4 + 8 × 6 + 5 × 10 = 48 + 48 + 50 = 146. Row 2: 7 × 4 + 10 × 6 + 9 × 10 = 28 + 60 + 90 = 178. So SP = [146; 178]: branch 1 took $146 and branch 2 took $178. Each entry is a sum of quantity-times-price terms, which is why the multiplication rule is built the way it is. The product PS is not defined: (3 × 1)(2 × 3) has inner dimensions 1 and 2, which do not match.

Multiplying network matrices. The square of the adjacency matrix counts two-step routes. (M²)_ij adds up, over every middle town k, the product (route i → k)(route k → j), which is 1 exactly when both roads exist. For the four towns: row 1 of M² is [0 0 1 2]. The 2 in position (1, 4) counts the two-step trips 1 → 2 → 4 and 1 → 3 → 4; the 1 in position (1, 3) counts 1 → 2 → 3. Row 2 is [0 0 0 1] for 2 → 3 → 4; rows 3 and 4 are all zeros. Multiplying the matrix does the counting that tracing the map would do by hand.

The misconception: multiply entry by entry. Because addition is entry by entry, students expect multiplication to be too. It is not; the entry-wise product is a different operation with no use here. Matrix multiplication is row-times-column, and that is what makes SP mean revenue and M² mean two-step routes. A second error is to multiply in the wrong order or without checking dimensions. Always write the dimensions beside each matrix, check that the inner pair matches, and read off the outer pair as the dimensions of the answer before computing a single entry.

About the numbers. Every price, fee, amount and business in this lesson's examples, tasks and questions is made up for practice; none describes a real company, product or offer.

Worked, adding and subtracting data matrices: Week one sales S has rows [12 8 5] and [7 10 9]; week two T has rows [15 6 9] and [5 12 8]. Both are 2 × 3, so the sum and difference are defined. S + T: rows [12 + 15, 8 + 6, 5 + 9] = [27 14 14] and [7 + 5, 10 + 12, 9 + 8] = [12 22 17]. T − S: rows [3 −2 4] and [−2 2 −1]. Branch 1 sold 3 more notebooks and 2 fewer pens in week two. Adding S to the 3 × 1 price column P is undefined: the dimensions differ.

Worked, scalar multiple of a payoff matrix: Payoff matrix G has rows [3 −1] and [0 2]. Doubling all the payoffs gives 2G with rows [6 −2] and [0 4]. Every entry, including the negative one, is multiplied by 2; the loss of 1 becomes a loss of 2. The dimensions stay 2 × 2. Halving the stakes would be 0.5G with rows [1.5 −0.5] and [0 1].

Worked, a product with matching dimensions: S is 2 × 3 with rows [12 8 5], [7 10 9]; P is 3 × 1, entries 4, 6, 10. Inner dimensions 3 and 3 match, so SP is 2 × 1. Entry 1: 12(4) + 8(6) + 5(10) = 48 + 48 + 50 = 146. Entry 2: 7(4) + 10(6) + 9(10) = 28 + 60 + 90 = 178. SP = [146; 178], the revenue in dollars at each branch. PS is undefined because (3 × 1)(2 × 3) has inner dimensions 1 ≠ 2.

Worked, a 2 × 2 times 2 × 2 product: A has rows [1 2] and [3 4]; B has rows [0 1] and [1 0]. AB is 2 × 2. Entry (1, 1): row 1 of A with column 1 of B: 1(0) + 2(1) = 2. Entry (1, 2): 1(1) + 2(0) = 1. Entry (2, 1): 3(0) + 4(1) = 4. Entry (2, 2): 3(1) + 4(0) = 3. AB has rows [2 1] and [4 3]: multiplying by B on the right swapped the columns of A.

Worked, two-step routes: For the four-town matrix M with rows [0 1 1 0], [0 0 1 1], [0 0 0 1], [0 0 0 0], compute entry (1, 4) of M²: row 1 of M with column 4 of M: 0(0) + 1(1) + 1(1) + 0(0) = 2. There are two two-step trips from town 1 to town 4, via town 2 and via town 3. Entry (1, 3): 0(0) + 1(1) + 1(0) + 0(0) = 1, the trip 1 → 2 → 3. Entry (2, 4): 0(0) + 0(1) + 1(1) + 1(0) = 1, the trip 2 → 3 → 4.

Recording task: Draw the four-town map with one-way roads 1 → 2, 1 → 3, 2 → 3, 2 → 4 and 3 → 4. Write its 4 × 4 adjacency matrix M with row and column headings, then compute all sixteen entries of M² by row-times-column, showing the four products added for at least three of the entries. Record M² as rows [0 0 1 2], [0 0 0 1], [0 0 0 0], [0 0 0 0], and next to each nonzero entry list the two-step routes it counts by tracing the map. Write one sentence on why row 4 of both M and M² is all zeros.

Independent performance: Invent a data situation of your own with at least two rows and three columns, such as hours worked by two people over three days, or points scored by three teams in two rounds, and write its matrix with headings. Then: (1) write a second matrix of the same dimensions for a later period and compute the sum and the difference, interpreting one negative entry; (2) choose a scalar with a meaning (a percentage increase, a doubling) and compute the scalar multiple; (3) build a column matrix of rates (pay per hour, price per item) whose dimensions make the product defined, compute the product, and interpret every entry; (4) write one product that is undefined and say which dimensions fail to match. Success criteria: the dimensions are written beside every matrix and the inner-outer rule is applied before each product; every product entry shows its row-times-column sum; the interpretation of each result uses the row and column meanings; and the undefined product is correctly identified with a reason.