Matrix Transpose Calculator
Select the matrix size and enter its elements to calculate its transpose step by step.
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Solved Exercises
The following are examples of problems solved by the calculator.
Exercise 1
Find the transpose of the matrix:
$$ \displaystyle A = \begin{bmatrix} 5 & -2 & 3 \\[0.5em] 0 & 1 & -4 \end{bmatrix} $$
Result
The transposed matrix is:
$$ \displaystyle A^T = \begin{bmatrix} 5 & 0 \\[0.5em] -2 & 1 \\[0.5em] 3 & -4 \end{bmatrix} $$
This matrix is obtained by swapping the rows and columns of the original matrix (e.g., row 1 of the original matrix becomes column 1 in the transposed matrix). We can see this process below.
Original matrix:
$$ \displaystyle A = \begin{bmatrix} \textcolor{#ef4444}{5} & \textcolor{#ef4444}{-2} & \textcolor{#ef4444}{3} \\[0.5em] \textcolor{#22c55e}{0} & \textcolor{#22c55e}{1} & \textcolor{#22c55e}{-4} \end{bmatrix} $$
Transposed matrix:
$$ \displaystyle A^T = \begin{bmatrix} \textcolor{#ef4444}{5} & \textcolor{#22c55e}{0} \\[0.5em] \textcolor{#ef4444}{-2} & \textcolor{#22c55e}{1} \\[0.5em] \textcolor{#ef4444}{3} & \textcolor{#22c55e}{-4} \end{bmatrix} $$
Exercise 2
Calculate the transpose of the matrix:
$$ \displaystyle A = \begin{bmatrix} 1 & 2 \\[0.5em] 3 & 4 \end{bmatrix} $$
Result
The transposed matrix is:
$$ \displaystyle A^T = \begin{bmatrix} 1 & 3 \\[0.5em] 2 & 4 \end{bmatrix} $$
The result is achieved by rearranging each row of the original matrix into its corresponding column position (for example, the first row becomes the first column). The procedure is shown below.
Original matrix:
$$ \displaystyle A = \begin{bmatrix} \textcolor{#ef4444}{1} & \textcolor{#ef4444}{2} \\[0.5em] \textcolor{#22c55e}{3} & \textcolor{#22c55e}{4} \end{bmatrix} $$
Transposed matrix:
$$ \displaystyle A^T = \begin{bmatrix} \textcolor{#ef4444}{1} & \textcolor{#22c55e}{3} \\[0.5em] \textcolor{#ef4444}{2} & \textcolor{#22c55e}{4} \end{bmatrix} $$
Exercise 3
Determine the transpose of the following matrix:
$$ \displaystyle A = \begin{bmatrix} 1 & 0 & -1 & 2 \\[0.5em] 3 & 4 & 0 & -2 \\[0.5em] -3 & 1 & 5 & 0 \end{bmatrix} $$
Result
The transposed matrix is:
$$ \displaystyle A^T = \begin{bmatrix} 1 & 3 & -3 \\[0.5em] 0 & 4 & 1 \\[0.5em] -1 & 0 & 5 \\[0.5em] 2 & -2 & 0 \end{bmatrix} $$
To obtain this matrix, the positions of the rows and columns are swapped (for example, the elements of the initial row 1 form the final column 1). This change is illustrated step by step below.
Original matrix:
$$ \displaystyle A = \begin{bmatrix} \textcolor{#ef4444}{1} & \textcolor{#ef4444}{0} & \textcolor{#ef4444}{-1} & \textcolor{#ef4444}{2} \\[0.5em] \textcolor{#22c55e}{3} & \textcolor{#22c55e}{4} & \textcolor{#22c55e}{0} & \textcolor{#22c55e}{-2} \\[0.5em] \textcolor{#f97316}{-3} & \textcolor{#f97316}{1} & \textcolor{#f97316}{5} & \textcolor{#f97316}{0} \end{bmatrix} $$
Transposed matrix:
$$ \displaystyle A^T = \begin{bmatrix} \textcolor{#ef4444}{1} & \textcolor{#22c55e}{3} & \textcolor{#f97316}{-3} \\[0.5em] \textcolor{#ef4444}{0} & \textcolor{#22c55e}{4} & \textcolor{#f97316}{1} \\[0.5em] \textcolor{#ef4444}{-1} & \textcolor{#22c55e}{0} & \textcolor{#f97316}{5} \\[0.5em] \textcolor{#ef4444}{2} & \textcolor{#22c55e}{-2} & \textcolor{#f97316}{0} \end{bmatrix} $$
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