Matrices

Maths · Class 12

Simulation · Maths · Class 12

Two moves, two orders

"Two moves, two orders" is a free, interactive Class 12 Maths simulation for JEE, built on the lesson "Order matters in a product" in Matrices. The orange tip is where (1, 0) lands and the blue tip is where (0, 1) lands. Together they are the two columns of M, and the whole unit square is carried along with them. A 2 × 2 matrix is a move of the plane, and a product is one move after another. Doing two moves in the other order usually lands somewhere else, which is why AB and BA can differ. JEE Main asked 28 questions from this chapter across the 42 shifts Lumi has analysed (2017–2026).

From the lesson Order matters in a product in Matrices. Change the values and watch what happens.

Two moves, two ordersMaths · Class 12

In JEE Main

42 shifts analysed

28 questions from Matrices across the 42 JEE Main shifts Lumi has analysed (2017–2026).

See the past questions

The idea behind it

NCERT §3.4.5

  • AB may be defined while BA is not. A 2 × 2 matrix times a 2 × 3 matrix is fine, but the 2 × 3 matrix times the 2 × 2 one is not, since 3 ≠ 2.
  • Both are defined exactly when A is m × n and B is n × m. Then AB is m × m and BA is n × n, so they can even have different orders: for a 2 × 3 A and 3 × 2 B, AB is 2 × 2 and BA is 3 × 3.
  • Even square matrices of the same order usually fail to commute. With A = [[1, 0], [0, −1]] and B = [[0, 1], [1, 0]], AB = [[0, 1], [−1, 0]] but BA = [[0, −1], [1, 0]].
  • Some pairs do commute: any two diagonal matrices of the same order, a matrix with I or O, and a matrix with its own powers. [[1, 0], [0, 2]] and [[3, 0], [0, 4]] give [[3, 0], [0, 8]] in either order.
  • A product of two non-zero matrices can be the zero matrix: [[0, −1], [0, 2]] times [[3, 5], [0, 0]] is O. So AB = O does not force A = O or B = O.
  • For the same reason AB = AC does not allow cancelling A to get B = C.

Questions

3 answers

What does the "Two moves, two orders" simulation show?
The orange tip is where (1, 0) lands and the blue tip is where (0, 1) lands. Together they are the two columns of M, and the whole unit square is carried along with 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 "Order matters in a product" in Matrices (Class 12 Maths).

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