Matrix Transpose Calculator
Transpose any matrix with a full element-by-element explanation. Every (i,j) entry becomes the (j,i) entry of Aᵀ.
Transpose any matrix instantly, with every element's row-to-column swap explained, plus automatic symmetry and dimension checks.
How Matrix Transpose Calculator Works
The transpose follows one simple rule: Aᵀ[j][i] = A[i][j]. Every row of the original matrix A becomes the matching column of the transposed matrix Aᵀ, effectively flipping the matrix over its main diagonal.
Because rows and columns swap, the dimensions flip too — an r×c matrix becomes a c×r matrix after transposing, so a non-square matrix actually changes shape (a 2×3 matrix becomes 3×2).
For square matrices, the calculator also checks two special properties: symmetric (A equals its own transpose, A = Aᵀ) and skew-symmetric (A = −Aᵀ), and lists the standard transpose identities — (Aᵀ)ᵀ = A, (A+B)ᵀ = Aᵀ+Bᵀ, and (AB)ᵀ = BᵀAᵀ.
See It In Action
Who Uses Matrix Transpose Calculator and Why
- Flipping a matrix's rows into columns to match the orientation a particular formula or downstream calculation requires.
- Checking whether a square matrix is symmetric (A = Aᵀ) or skew-symmetric (A = −Aᵀ).
- Verifying transpose identities, like (AB)ᵀ = BᵀAᵀ, for a linear algebra homework problem.
- Confirming how a non-square matrix's dimensions change (rows and columns swap) after transposing.
Mistakes to Avoid
- Assuming transposing changes the values inside a matrix — it only relocates each entry's position (Aᵀ[j][i] = A[i][j]), it never changes what the values themselves are.
- Forgetting that a non-square matrix's dimensions flip after transposing — a 2×3 matrix becomes a 3×2 matrix, not another 2×3 matrix.
- Mixing up (AB)ᵀ with AᵀBᵀ — the transpose of a product reverses the multiplication order, giving BᵀAᵀ instead, one of the most commonly confused matrix identities.
Tips for Best Results
- Use the automatic symmetric/skew-symmetric check as a shortcut instead of manually comparing every entry against its mirror position.
- Remember (Aᵀ)ᵀ always returns the original matrix A, which is a quick way to sanity-check that a transpose was computed correctly.
Fixing Common Problems
My transposed matrix has a different shape than I expected. — That's correct behavior for a non-square matrix — an r×c matrix always becomes a c×r matrix after transposing.
I'm not sure if my matrix is symmetric. — Check the calculator's automatic symmetry check — a square matrix is symmetric only if it equals its own transpose exactly, entry for entry.
Terms Explained
Symmetric matrix: A square matrix that equals its own transpose (A = Aᵀ), where entries mirror across the main diagonal.
Skew-symmetric matrix: A square matrix that equals the negative of its own transpose (A = −Aᵀ).