Calculate matrix transpose, check symmetry, and verify transpose properties.
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The transpose A^T is obtained by swapping rows and columns: (A^T)i,j = Aj,i. If A is m×n, then A^T is n×m. Key rules: (A^T)^T = A, (A+B)^T = A^T + B^T, and (AB)^T = B^T A^T.
A matrix is symmetric when A = A^T and skew‑symmetric when A = −A^T (all diagonal entries are zero). This tool flags these properties automatically.
For square matrices, det(A^T) = det(A) and rank(A^T) = rank(A). Transpose preserves determinant magnitude and rank.