Transpose of a Product
The transpose of a product reverses the order.
Formula
Variables
Example
Order flips on transpose
How it works
For matrix products, transposition swaps rows and columns, so the entries that end up in position (i, j) after transposing AB come from position (j, i) in AB. That reversal forces the factors to switch order: the jth column of A and ith row of B become the ith row of B^T and jth column of A^T, which matches B^T A^T. This identity is a structural fact about how matrix multiplication is built from dot products, not just a computational trick.
When to use it
Use this identity whenever you need to simplify or rearrange transposes of products, especially in proofs, matrix algebra manipulations, and derivations involving symmetric matrices or bilinear forms. It is also useful in statistics, optimization, and applied linear algebra when converting expressions like (X^T X)^T or checking symmetry.
Worked example, step by step
Let A = [[1, 2], [3, 4]] and B = [[0, 5], [6, 7]]. First compute AB: AB = [[1·0 + 2·6, 1·5 + 2·7], [3·0 + 4·6, 3·5 + 4·7]] AB = [[12, 19], [24, 43]] Now transpose AB: (AB)^T = [[12, 24], [19, 43]] Next compute B^T and A^T: B^T = [[0, 6], [5, 7]] A^T = [[1, 3], [2, 4]] Multiply B^T A^T: B^T A^T = [[0·1 + 6·2, 0·3 + 6·4], [5·1 + 7·2, 5·3 + 7·4]] B^T A^T = [[12, 24], [19, 43]] So (AB)^T = B^T A^T = [[12, 24], [19, 43]].
Common mistakes to avoid
- Assuming (AB)^T = A^T B^T; the order must reverse.
- Forgetting that matrix dimensions must still be compatible after transposing each factor.
- Applying the rule to sums or scalar multiples incorrectly without using the separate transpose properties first.
Did You Know?
Transposing swaps rows and columns, and a matrix equal to its own transpose is called symmetric.
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