AB means: apply B first, then A.
Multiplying matrices chains two moves into one. (AB)v = A(Bv): B moves the plane first, then A moves the result.
Think: socks and shoes
Socks first, then shoes. Swap the order and you get a very different result. Matrix moves work the same way.
Read a product from right to left, like f(g(x)) in maths. In AB, B acts first.
Column by column
Where does (1, 0) end up after both moves? B sends it to b₁, the first column of B. Then A sends that to Ab₁. So the first column of AB is A times the first column of B. Same for the second column.
You already know matrix times vector. Matrix times matrix is just that, once per column.
Worked example
The rules
- AB is usually not BA. Order matters.
- (AB)C = A(BC). Either way, C acts first, then B, then A. Brackets only change how you write it.
- det(AB) = det(A)·det(B). If B doubles areas and A triples them, together they multiply areas by 6 (chapter 5).
Where you meet this
A robot arm is a chain of rotations, one per joint. To find the gripper, multiply them in joint order. Swap two and the gripper ends up somewhere else.
Worked examples
Example 1 · Two orders
A = 1201 (shear), B = 2001 (stretch).
Stretch then shear differs from shear then stretch.
Example 2 · Two quarter turns
R = 0−110 turns by 90°.
Two quarter turns make a half turn. For rotations, multiplying adds the angles.
Problems
Try each one before you open the solution.
Compute AB and BA for A = 1110 and B = 0110.
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AB = 1101 and BA = 1011. Different.
A = 2003, B = 1401. Find det(AB) without multiplying.
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det A = 6 and det B = 1, so det(AB) = 6. Check: AB = 2803, det = 6 ✓
Find only the first column of AB for A = 1234 and B = 1011.
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Column 1 of AB = A times column 1 of B = A·(1, 1) = (1 + 2, 3 + 4) = (3, 7).