Matrix Transpose Calculator – Rows to Columns

Transposing reflects a matrix across its main diagonal: entry (i, j) moves to (j, i). The 2×3 matrix [[1,2,3],[4,5,6]] becomes the 3×2 matrix [[1,4],[2,5],[3,6]] — row one turns into column one. The calculator transposes any m×n input up to 8×8, shows the entry-by-entry mapping, and verifies the algebra identities: applying transpose twice restores the original, (Aᵀ)ᵀ = A; sums transpose termwise; and products reverse order, (AB)ᵀ = BᵀAᵀ — demonstrated numerically on your own matrices. Special shapes have names: a matrix equal to its transpose is symmetric ([[2,1],[1,3]]); one satisfying A = −Aᵀ is skew-symmetric with zero diagonal. The dot product hides here too: u·v = uᵀv, which is why statistics and regression lean on transposes constantly.

1. 1

   Enter your matrix

   Any dimensions up to 8×8

2. 2

   Reflect across the diagonal

   Each (i, j) entry is mapped to (j, i)

3. 3

   Read the transposed matrix

   Rows have become columns and vice versa

4. 4

   Check the identities

   (Aᵀ)ᵀ and reversed products are computed as proof

• 1

  Transpose of m×n is n×m — dimensions always swap

• 2

  (AB)ᵀ = BᵀAᵀ — order flips

• 3

  Symmetric ⇔ A = Aᵀ

• Transposing only the outer rows rather than reflecting the whole diagonal

• Writing (AB)ᵀ = AᵀBᵀ — the order must reverse

• Assuming every square matrix commutes with its transpose

• Forgetting vectors transpose between row and column shapes