Inverse of a Product: The "Socks and Shoes" Principle
You put on your socks first, then your shoes. To take them off, you can't remove the socks while the shoes are still on — you must reverse the order: shoes off first, then socks.
That's exactly the inverse of a product of matrices. If you apply transformation A first, then B, the combined effect is BA (read right-to-left: A acts first, then B). To undo it, undo B first, then A:
(AB)−1=B−1A−1
The order flips — forced by the logic of undoing.
Why the order must reverse
Check that B−1A−1 is the inverse of AB. We need (AB)(B−1A−1)=I and (B−1A−1)(AB)=I:
(AB)(B−1A−1)=A(BB−1)A−1=AIA−1=AA−1=I
B and B−1 cancel first, leaving A and A−1 to cancel. The other check works the same way:
(B−1A−1)(AB)=B−1(A−1A)B=B−1IB=B−1B=I
If you tried (AB)−1=A−1B−1 instead:
(AB)(A−1B−1)=A(BA−1)B−1
and BA−1 is not I — the matrices are in the wrong order. So the reversal is essential.
A common mistake is writing (AB)−1=A−1B−1. This is false unless A and B commute (which they almost never do). Always flip the order.
A concrete example with numbers
Let A=(1021) and B=(1101), with inverses:
A−1=(10−21),B−1=(1−101)
Then:
AB=(1021)(1101)=(3121),(AB)−1=(1−1−23)
Now compute B−1A−1:
B−1A−1=(1−101)(10−21)=(1−1−23)
They match. Try A−1B−1 and you'll get a different matrix — the wrong answer.
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