Matrix Transpose Calculator

What is a Matrix Transpose?

The transpose of a matrix is one of the simplest operations in linear algebra. When you transpose a matrix, you rotate it across its main diagonal, effectively flipping the grid. The rows of the original matrix become the columns of the transposed matrix, and the columns of the original matrix become the rows of the transposed matrix.

If A is an m x n matrix, then its transpose, denoted A^T, is an n x m matrix. Formally, we define the elements of A^T as:

A^T_i,j = A_j,i

Worked Example of a Matrix Transpose

Let's find the transpose of a 2x3 matrix A:

By swapping rows and columns, the first row [1, 2, 3] becomes the first column, and the second row [4, 5, 6] becomes the second column. The resulting 3x2 transposed matrix A^T is:

Properties of the Transpose Operation

The transpose operation satisfies several algebraic properties that are useful in matrix proofs and computational simplifications:

• (A^T)^T = A: Transposing a matrix twice returns it to its original state.
• (A + B)^T = A^T + B^T: The transpose of a sum is equal to the sum of the transposes.
• (AB)^-1 = (A^-1)^T: The transpose of a product of two matrices reverses their order, which is a critical identity: (AB)^T = B^TA^T.
• (cA)^T = c(A^T): Scaling a matrix and then transposing it is the same as transposing it first and then scaling.

If a square matrix is equal to its own transpose (A = A^T), the matrix is called symmetric. If a matrix is equal to the negative of its transpose (A = -A^T), it is called skew-symmetric.