Row by column
"Row by column" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Multiplying row by column" in Matrices. The left factor sits to the left of the product and the right factor sits above it. The entry you pick lights up its row on the left and its column above. Multiply the matching pairs and add them. Entry (i, k) of a product is row i of the left factor run against column k of the right factor. Swapping the factors changes which rows meet which columns, and here even the size of the answer. JEE Main asked 28 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
From the lesson Multiplying row by column in Matrices. Change the values and watch what happens.
In JEE Main
42 shifts analysed
28 questions from Matrices across the 42 JEE Main shifts Lumi has analysed (2017–2026).
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.
Questions
3 answers
- What does the "Row by column" simulation show?
- The left factor sits to the left of the product and the right factor sits above it. The entry you pick lights up its row on the left and its column above. Multiply the matching pairs and add them.
- Does JEE Main ask questions from Matrices?
- Yes. JEE Main asked 28 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).
- Is this simulation free?
- Yes. It is free and runs in the browser on a phone or a laptop. It belongs to the lesson "Multiplying row by column" in Matrices (Class 12 Maths).
Embed this simulation
Free, with a link back
Teaching this idea? Put the simulation on your own page or LMS with this code.