Flip a matrix over its diagonal, switching rows and columns index (i,j) → (j,i).
Input Matrix (A)
[0,0]
[0,1]
[0,2]
[1,0]
[1,1]
[1,2]
[2,0]
[2,1]
[2,2]
Transposed Matrix (Aᵀ)
"The reflection across the main diagonal."
1[0,0]
0[0,1]
0[0,2]
0[1,0]
1[1,1]
0[1,2]
0[2,0]
0[2,1]
1[2,2]
New Dimensions3 × 3
Elements9
About this calculator
What is a Matrix Transpose?
The transpose of a matrix is an operator which flips a matrix over its main diagonal. Mathematically, it switches the row and column indices of the matrix A to produce matrix Aᵀ. If A is an m × n matrix, Aᵀ will be an n × m matrix.
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Pro Tips
The transpose of a transpose is the original matrix: (Aᵀ)ᵀ = A.
The transpose of a sum is the sum of transposes: (A + B)ᵀ = Aᵀ + Bᵀ.
For product: (AB)ᵀ = BᵀAᵀ. This is known as the 'Reversal Rule'.
The determinant of a matrix is equal to the determinant of its transpose: det(A) = det(Aᵀ).
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Fun Facts
"A square matrix where A = Aᵀ is called a Symmetric Matrix. Many physical systems (like springs) are modeled by symmetric matrices."
"If A = -Aᵀ, the matrix is Skew-Symmetric. All diagonal elements of a skew-symmetric matrix must be zero."
"Transposing is used heavily in Computer Graphics to rotate and transform 3D models."
"In data science, transposing is the first step in calculating 'Correlation Matrices'."
