Transpose of a matrix
The transpose A′ (or Aᵀ) turns rows into columns. An m × n matrix becomes n × m, and products reverse: (AB)′ = B′A′.
What transposing does
The transpose of A, written A′ or Aᵀ, is the matrix you get by turning each row of A into a column, in the same order. If A is m × n, then A′ is n × m. For example, if A = [1, 2, 3; 4, 5, 6] (order 2 × 3), then A′ = [1, 4; 2, 5; 3, 6] (order 3 × 2). The element in row i, column j of A′ is the element in row j, column i of A. The entries on the main diagonal of a square matrix stay where they are. (Matrices are written row by row here, with rows separated by semicolons.)
Properties of transpose (orders suitable)
| Property | Statement |
|---|---|
| Transpose of a transpose | (A′)′ = A |
| Scalar multiple | (kA)′ = kA′ for any scalar k |
| Sum | (A + B)′ = A′ + B′ |
| Difference | (A − B)′ = A′ − B′ |
| Product (reversal law) | (AB)′ = B′A′, and for three matrices (ABC)′ = C′B′A′ |
| Identity | I′ = I |
Example 1: verify the reversal law
If A = [1, 2; 0, 3] and B = [2, −1; 1, 4], verify that (AB)′ = B′A′. Also show that (AB)′ ≠ A′B′.
- Rule: multiply row by column, then turn rows into columns.
- AB = [(1)(2) + (2)(1), (1)(−1) + (2)(4); (0)(2) + (3)(1), (0)(−1) + (3)(4)] = [4, 7; 3, 12]
- (AB)′ = [4, 3; 7, 12]
- B′ = [2, 1; −1, 4] and A′ = [1, 0; 2, 3]
- B′A′ = [(2)(1) + (1)(2), (2)(0) + (1)(3); (−1)(1) + (4)(2), (−1)(0) + (4)(3)] = [4, 3; 7, 12]
- A′B′ = [(1)(2) + (0)(−1), (1)(1) + (0)(4); (2)(2) + (3)(−1), (2)(1) + (3)(4)] = [2, 1; 1, 14]
(AB)′ = B′A′ = [4, 3; 7, 12], but A′B′ = [2, 1; 1, 14] is different.
Example 2: use the properties
If A = [2, −1; 3, 0] and B = [1, 4; −2, 5], find (2A − B)′ in two ways.
- Way 1: 2A − B = [4 − 1, −2 − 4; 6 − (−2), 0 − 5] = [3, −6; 8, −5].
- So (2A − B)′ = [3, 8; −6, −5].
- Way 2, by the properties: (2A − B)′ = 2A′ − B′.
- A′ = [2, 3; −1, 0], so 2A′ = [4, 6; −2, 0]. Also B′ = [1, −2; 4, 5].
- 2A′ − B′ = [4 − 1, 6 − (−2); −2 − 4, 0 − 5] = [3, 8; −6, −5].
(2A − B)′ = [3, 8; −6, −5], the same both ways.
In a spreadsheet, 'Paste Special, Transpose' swaps rows and columns, turning a table of students against subjects into subjects against students. That is exactly A′.
1 marks: “If A is a 3 × 5 matrix, what is the order of A′?” 5 × 3
2 marks: “If A′ = [2, −1; 0, 4] and B = [1, 3; −2, 0], find (A + B)′.” (A + B)′ = A′ + B′. B′ = [1, −2; 3, 0], so (A + B)′ = [2 + 1, −1 − 2; 0 + 3, 4 + 0] = [3, −3; 3, 4].
3 marks: “If A = [1, −1; 2, 3] and B = [2, 0; 1, 4], verify that (AB)′ = B′A′.” AB = [1, −4; 7, 12], so (AB)′ = [1, 7; −4, 12]. B′ = [2, 1; 0, 4] and A′ = [1, 2; −1, 3], so B′A′ = [1, 7; −4, 12]. Both sides match.
Transpose flips rows into columns; in a product it also flips the order: (AB)′ = B′A′, just as you take off your shoes before your socks.