Sec 3 E-Math Matrices notes

A matrix is a rectangular array of numbers. A matrix with 2 rows and 3 columns has order 2 × 3 — rows first, then columns, always. Individual entries are addressed as aᵢⱼ, where i is the row and j the column: in [[2, 5], [1, 3]], the entry a₁₂ (row 1, column 2) is 5, while a₂₁ (row 2, column 1) is 1.

• Row matrix: a single row, order 1 × n, such as [3, 5, 7].
• Column matrix: a single column, order m × 1, such as [[4], [2]].
• Square matrix: same number of rows and columns; its diagonal runs from top-left to bottom-right.
• Zero matrix: every entry 0; identity matrix I (square): 1s on the diagonal and 0s elsewhere.

Two matrices add (or subtract) only when they have the same order: add (or subtract) matching entries. A scalar multiple kA multiplies every entry of A by the number k. Both operations keep the original order.

Addition is commutative (A + B = B + A) and associative; subtraction is neither, so A − B ≠ B − A in general — flip every sign of the second matrix and you get the other order.

The product AB pairs each row of A with each column of B: multiply the matching entries, then add. The entry in row i, column j of AB uses row i of A and column j of B. Before computing, check the dimension rule: AB exists only when the number of columns of A equals the number of rows of B, and (AB) then has order (rows of A) × (columns of B).

1. 1Check dimensions

   A = [[2, 5], [1, 3]] is 2 × 2 and B = [[4, 1], [2, 2]] is 2 × 2: inner numbers match (2 = 2), so AB is defined, with order 2 × 2.

2. 2Row 1 × Column 1

   (2)(4) + (5)(2) = 8 + 10 = 18 — top-left entry.

3. 3Row 1 × Column 2

   (2)(1) + (5)(2) = 2 + 10 = 12 — top-right entry.

4. 4Row 2 across both columns

   (1)(4) + (3)(2) = 10 and (1)(1) + (3)(2) = 7. So AB = [[18, 12], [10, 7]].

Special matrices behave nicely: multiplying by the identity I leaves A unchanged (AI = IA = A), and multiplying by a zero matrix (where defined) gives a zero matrix. Note that AB can be defined while BA is not: for A of order 2 × 3 and B of order 3 × 4, AB is 2 × 4 but BA does not exist, since 4 ≠ 2.

The syllabus emphasises interpreting matrices: real data — shops against products, weeks against items, students against subjects — lives naturally in rows and columns. Decode the table before computing: state what a row means, what a column means, and therefore what any entry means. A 2 × 3 matrix of two branches and three products puts branch 2’s sales of product 1 in position (2, 1).

When two data matrices describe different aspects — quantities sold and prices per item — their product combines them: quantity × price lands every total in one small matrix. That is the engine of the applied questions in this chapter.

Applied matrix questions follow one pattern: a quantity matrix (rows and columns labelled by shops and items) multiplies a price column matrix, producing a column of money totals — one per shop. The dimension rule guides the set-up: quantities of order 2 × 3 meet prices of order 3 × 1, giving totals of order 2 × 1.

1. 1Label everything

   Stock S = [[10, 5, 8], [6, 9, 4]] for two branches × three items; restock R = [[4, 2, 1], [3, 1, 5]].

2. 2Add for the full stock

   S + R = [[14, 7, 9], [9, 10, 9]].

3. 3Price it

   Prices [[2], [3], [1]] per item: (S + R) × prices multiplies each branch’s row into the price column.

4. 4Read the totals

   Branch 1: 14(2) + 7(3) + 9(1) = 58; Branch 2: 9(2) + 10(3) + 9(1) = 57. Branch 1’s stock is worth $58, Branch 2’s $57.

Check dimensional sense in word problems too: multiplying a 1 × 2 price row by a 2 × 1 quantity column gives one grand total (1 × 1), while the reverse order gives a 2 × 2 table of pairwise products — useful for a breakdown, but not the same question. Decide what answer shape you need, then order the factors to produce it.