Transpose of a matrix

On this post, we explain how to find the transpose of a matrix. You will also see solved problems so that you have no doubts about how to transpose a matrix. And finally, we will show you all the properties of the transpose of a matrix.

How to find the transpose matrix?

To find the transpose of a matrix, exchange the rows of the matrix for its columns, that is, the rows of the transposed matrix are the columns of the original matrix and the columns of the transposed matrix are the rows of the original matrix.

Thus, the transpose of a matrix is the matrix obtained by switching the rows of the matrix by its columns. Moreover, the transpose of a matrix is indicated by writing a “T” at the top right of the matrix (AT).

Example of the transpose of a matrix

Once we’ve seen the definition of the transpose of a matrix, let’s see an example of how to calculate the transpose of a matrix. We are going to transpose the following matrix:

 \displaystyle A= \begin{pmatrix} 2 & 3 & 1 \\[1.1ex] 4 & 5 & 0 \end{pmatrix}

To transpose matrix A we just have to interchange its rows for its columns. So, the first row of the matrix becomes the first column of the matrix, and the second row of the matrix becomes the second column of the matrix:

 \displaystyle A^T= \begin{pmatrix} 2 & 4 \\[1.1ex] 3 & 5 \\[1.1ex] 1 & 0 \end{pmatrix}

Logically, the dimension of a matrix changes when it is transposed. In this case, matrix A was a 2×3 dimension matrix, and its transpose is a 3×2 dimension matrix.

As you can see, there is no formula to find the transpose of a matrix, but transposing a matrix is not very complicated: you just have to flip the matrix over its diagonal.

Practice problems on transpose of a matrix

Below are several solved problems so that you can practice how to transpose a matrix.

Problem 1

Transpose the following 2×2 matrix:

\displaystyle A= \begin{pmatrix} 1 & 5\\[1.1ex] 7 & 2 \end{pmatrix}

\displaystyle A^T= \begin{pmatrix} 1 & 7\\[1.1ex] 5 & 2 \end{pmatrix}

 

Problem 2

Find the transpose of the following 3×3 matrix:

\displaystyle B= \begin{pmatrix} -1 & 4 & 3 \\[1.1ex] 5 & 3 & 2 \\[1.1ex] 6 & 0 & 9 \end{pmatrix}

 \displaystyle B^T= \begin{pmatrix} -1 & 5 & 6 \\[1.1ex] 4 & 3 & 0 \\[1.1ex] 3 & 2 & 9 \end{pmatrix}

 

Problem 3

Transpose the following matrix formed by one single row:

\displaystyle C= \begin{pmatrix} 2 & 6 & -1 \end{pmatrix}

\displaystyle C^T= \begin{pmatrix} 2 \\[1.1ex] 6 \\[1.1ex] -1 \end{pmatrix}

The transpose of a row matrix is always a column matrix. And vice versa, transposing a column matrix always results in a row matrix.

 

Problem 4

Calculate the transpose of the following rectangular matrix:

 \displaystyle D= \begin{pmatrix} 9 & 0 \\[1.1ex] 2 & -1 \\[1.1ex] 5 & 3 \end{pmatrix}

 \displaystyle D^T= \begin{pmatrix} 9 & 2 & 5 \\[1.1ex] 0 & -1 & 3 \end{pmatrix}

 

Problem 5

Transpose the following square 4×4 matrix:

\displaystyle E= \begin{pmatrix} 0&5&6&-2\\[1.1ex]4&-1&9&0\\[1.1ex]8&2&-2&11\\[1.1ex]7&-3&4&3\end{pmatrix}

\displaystyle E^T= \begin{pmatrix}0&4&8&7\\[1.1ex]5&-1&2&-3\\[1.1ex]6&9&-2&4\\[1.1ex]-2&0&11&3\end{pmatrix}

 

Properties of the transpose of a matrix

The transpose of a matrix has the following characteristics:

  • Involutory property: The transpose of a transposed matrix is equal to the original matrix.

 \left(A^T\right)^T = A

  • Distributive property: adding two matrices and then transposing the result is the same as transposing each matrix first and then adding them:

 \left(A+B\right)^T=A^T+B^T

See: matrix addition

  • Linear property (product of transposed matrices): multiplying two matrices and then transposing the result is equivalent to transposing each matrix first and then multiplying them but changing their order of multiplication:

\left(A\cdot B\right)^T=B^T\cdot A^T

See: matrix multiplication

  • Linear Property (scalar): transposing the result of the product of a matrix by a scalar is the same as multiplying the already transposed matrix by the scalar.

\left(c\cdot A\right)^T=c\cdot A^T

  • Determinant of a transposed matrix: the determinant of a matrix equals to the determinant of its transpose.

det(A)=det(A^T)

  • Inverse of a transposed matrix: calculating the inverse of a transposed matrix is the same as first calculating its inverse and then transposing the result.

\left(A^T\right)^{-1}=\left(A^{-1}\right)^T

Thus, if a matrix is invertible, the transpose of that matrix is also invertible.

  • Symmetric matrix: when the transpose of a matrix results in the same matrix, this matrix is called symmetric matrix.

 A^T=A

  • Antisymmetric matrix: when the transpose of a matrix results in the same matrix but with all elements changed sign, it is an antisymmetric matrix, also know as skew-symmetric matrix.

 A^T=-A

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