Matrix Transpose Calculator - 2x2 Row to Column Swap

A matrix transpose calculator swaps the rows and columns of a 2x2 input matrix A and returns the matching transposed matrix A^T in the result panel. Type the four cells of A in row order, and the calculator shows the four cells of A^T with the same numbers in their new positions.

• • Linear algebra homework: Confirm a 2x2 transpose by hand and check that the off-diagonal cells traded places without redoing the swap.
• • Statistics and covariance matrices: Transpose a small 2x2 covariance or correlation slice so columns become rows for a print-ready table.
• • Computer graphics transforms: Swap the rows and columns of a 2x2 rotation or scale matrix to study its transpose, which inverts the rotation direction.
• • Robotics and kinematics: Work through 2x2 coordinate transform transposes in introductory statics and confirm the result by hand.

The transpose is one of the first operations introduced after addition and scalar multiplication in any linear algebra course, and it is the building block for symmetric matrices, the adjugate used in the inverse formula, and the change-of-basis step that shows up in eigenvalue problems. Both shapes of A and A^T are 2x2 for this calculator, so the four cells of A become the four cells of A^T in their new positions.

The swap is the entire operation: the top-left and bottom-right cells of A stay where they are in A^T, and the top-right and bottom-left cells trade places. The diagonal entries are unchanged, and the off-diagonal entries are flipped across the main diagonal.

According to Khan Academy, the transpose of a matrix shows up in linear transformations, where a matrix acting on a column vector by multiplication corresponds to the transposed matrix acting on a row vector

The calculator reads the four cells of the input matrix A, then places each cell into its transposed position: the first row of A becomes the first column of A^T, and the second row of A becomes the second column of A^T.

• A[i][j]: The entry in row i, column j of the 2x2 input matrix A.
• A^T[i][j]: The entry in row i, column j of the transposed 2x2 matrix A^T.
• m x n: The shape of A. For this calculator m = 2 and n = 2, so the transposed matrix A^T is also 2x2.

The transpose is purely a rearrangement of cells; no arithmetic happens inside the cells. Each cell of A is copied into exactly one cell of A^T, so the multiset of values is identical, only the positions change.

Because the swap only moves cells around, the transpose keeps every sign, every decimal, and every integer that was in the input. The calculation runs in full double precision and rounds only the display to four decimal places.

According to Wikipedia, the transpose of a matrix A is the matrix A^T whose rows are the columns of A, so the entry at row i, column j of A^T equals the entry at row j, column i of A

When the same four cells are summed with another 2x2 matrix instead of being swapped, the matrix addition calculator returns the matching cell-by-cell sum.

Four short ideas cover the cell-swap rule, the symmetric case, and the shapes that show up when the transpose is used in larger formulas.

These four ideas are the backbone of the transpose, and they reappear in the adjugate formula, in the symmetric and skew-symmetric matrix definitions, and in the dot product, where (A x) · y = x · (A^T y).

Symmetric matrices turn the transpose into a no-op, skew-symmetric matrices turn the transpose into a sign flip, and rectangular matrices swap rows and columns so the cell-by-cell rule from the 2x2 case still applies one row or column at a time.

For the inverse step that often uses A^T inside the adjugate formula, the matrix inverse calculator walks through the 2x2 inverse from the same four cells.

Type the four cells of the 2x2 input matrix A in row order, then read the four transposed cells of A^T in the result panel. The output updates as you type.

1. 1 Type the first row of matrix A: Fill in the A row 1, col 1 and A row 1, col 2 cells with the top row of your input matrix.
2. 2 Type the second row of matrix A: Fill in the A row 2, col 1 and A row 2, col 2 cells with the bottom row of your input matrix.
3. 3 Read A^T row 1 (the first row of A becomes the first column of A^T): The first output row shows A11 unchanged and the new top-right cell A21, which was the bottom-left of A.
4. 4 Read A^T row 2 (the second row of A becomes the second column of A^T): The second output row shows the new middle-left cell A12, which was the top-right of A, followed by A22 unchanged.
5. 5 Reset or change the inputs: Use Reset to restore the default 2x2 matrix, or change any cell to recompute the transpose automatically.

Suppose you are working through a 2D graphics pipeline and need to transpose the 2x2 matrix A = [[2, 1], [0, 3]]. Type the entries of A in the four input cells, and the result panel gives you A^T = [[2, 0], [1, 3]] for the swap.

To multiply the transposed matrix by another matrix with the row-times-column rule, the matrix multiplication calculator follows the same cell-by-cell dot product on small matrices.

The transpose rule is short, but mixing up the row-column order is the most common source of errors. The calculator keeps both the input and the transposed cells on screen at the same time.

• • Cell-by-cell swap without redoing the bookkeeping: Each of the four output cells is labeled with its source cell from A, so you read the transposed value directly from the matching input.
• • Input matrix and transpose stay on screen together: The A cells remain visible in the form above the result panel, so you can cross-check the original values against the transposed cells.
• • Matches the textbook transpose rule exactly: A^T[i][j] = A[j][i] is the rule from the first chapter of any linear algebra text, and the calculator applies it the same way for decimals, negatives, and zero.
• • Works for positive, negative, and decimal entries: Cells can be 0.25, 1.5, -3, or any other real number; the calculation keeps full double precision internally and rounds only the display.

For a problem that mixes the transpose with other matrix operations on the same 2x2 input, the related operations on small matrices are covered by the same family of calculators. The page-level links below keep the workflow moving once A^T is in hand.

If you need to scale the transposed matrix A^T by a single scalar k instead of swapping cells, the matrix by scalar calculator returns the matching kA^T result.

A handful of input choices and structural facts decide whether A^T is what you expect.

• • The calculator is limited to 2x2 inputs. For 3x3 and larger matrices, the same row-to-column rule applies, but you would need a 3x3 tool to enter them all.
• • Output values are rounded to four decimal places for display, so a true fraction like 1/3 will appear as 0.3333 even though the cell of A may have been written as 1/3 in the source data.

If the matrix is part of a 2D or 3D transformation, the cells are usually integers or short decimals, and the transposed matrix is the same shape with the same multiset of values. The transpose only reorders the cells, it does not change them.

As published by Wolfram MathWorld, the transpose swaps the row and column indices of every entry, and a square matrix is symmetric exactly when it equals its own transpose

If you only need the diagonal sum of the transposed matrix, the matrix trace calculator shows the same total as the trace of the original input.