ConverterHub

2x2 Matrix Transpose Calculator

Enter the four entries of a 2×2 matrix to transpose it.

How it is calculated

The matrix is entered as [[a, b], [c, d]]: a and b are the first row, c and d the second. The transpose swaps rows and columns, giving [[a, c], [b, d]]. The diagonal entries a and d stay in place and only b and c change places. The tool prints the new first row as a, c and the new second row as b, d. All four entries are required and may be negative or have decimals; a missing or non-numeric entry gives no result. No arithmetic is done on the values, so the transpose is exact apart from display. Entries show up to four decimals with trailing zeros removed, but a small non-zero entry gets extra decimals (up to 12) so it never appears as 0. Large whole numbers use thousands separators.

Worked example

Take the matrix with first row 2, −5 and second row 7, 0.5. So a = 2, b = −5, c = 7, d = 0.5. The diagonal entries 2 and 0.5 stay where they are. Swap b and c. The new first row is a, c = 2, 7. The new second row is b, d = −5, 0.5. The tool shows row 1 as 2, 7 and row 2 as -5, 0.5, which is the transpose. Transposing it again gives the original matrix back.

Try this example →

Common mistakes

  • Entering the matrix by columns instead of rows. The fields are a, b for the first row and c, d for the second; column-wise entry gives the transpose of what you meant, which is the original matrix.
  • Reading the output rows as the columns of the input. The two output lines are the rows of the transposed matrix, so the first output line is a, c, not a, b.
  • Expecting the diagonal to change. Only the off-diagonal entries b and c swap; if b equals c, the matrix is symmetric and the output looks the same as the input.
  • Entering a fraction like 1/2. The fields take decimals, so type 0.5.
  • Using the result for a larger matrix by dropping entries. This tool handles only 2×2; a 3×3 matrix cannot be entered.

About this calculator

The transpose of a matrix, written Aᵀ, is formed by swapping its rows and columns: the entry in row 1, column 2 moves to row 2, column 1, and vice versa, while the diagonal entries (row 1, column 1 and row 2, column 2) stay exactly where they are. For a 2×2 matrix this simply means swapping the two off-diagonal entries.

Transposition is one of the most basic operations in linear algebra, used to convert a row vector into a column vector and back, to test whether a matrix is symmetric (a matrix equals its own transpose, A = Aᵀ), and as a building block in more advanced operations like computing a matrix's inverse or solving least-squares problems. In practical terms it shows up in computer graphics when switching between row-major and column-major conventions for transformation matrices, and in data science when reshaping a small feature matrix.

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