Simulation · Maths · Class 12

Row by column

From the lesson Multiplying row by column in Matrices. Change the values and watch what happens.

The idea behind it

NCERT §3.4.5

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 numbers must match and the outer numbers give the order.
The entry of AB in row i and column k is found by running along row i of A and down column k of B, multiplying matching entries and adding: cᵢₖ = aᵢ₁b₁ₖ + aᵢ₂b₂ₖ + … + aᵢₙbₙₖ.
Shopping example: two buyers want (2 pens, 5 books) and (8 pens, 10 books). At prices 5 and 50 rupees they pay 2·5 + 5·50 = 260 and 8·5 + 10·50 = 540; at prices 4 and 40 they pay 208 and 432. As one product, [[2, 5], [8, 10]] times [[5, 4], [50, 40]] is [[260, 208], [540, 432]].
Worked example: [[6, 9], [2, 3]] (2 × 2) times [[2, 6, 0], [7, 9, 8]] (2 × 3) is the 2 × 3 matrix [[75, 117, 72], [25, 39, 24]]; for instance 6·2 + 9·7 = 75.
Worked example: [[1, −1, 2], [0, 3, 4]] (2 × 3) times the 3 × 2 matrix with rows (2, 7), (−1, 1), (5, −4) is [[13, −2], [17, −13]].
Worked example: costs per contact of 40, 100 and 50 paise, times contacts (1000, 500, 5000) in one city and (3000, 1000, 10000) in another, give 340000 and 720000 paise, that is ₹3400 and ₹7200.
A 2 × 2 matrix times a column (x, y) gives a new point. The columns of the matrix are where (1, 0) and (0, 1) land, so a matrix can be read as a move of the plane: stretch, shear, rotation or reflection.

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