Transpose of a Matrix

Transpose of a matrix is a very common method used for matrix transformation in linear algebra. Transpose of a matrix is obtained by interchanging the rows and columns of the given matrix or vice versa. Transpose of a matrix can be utilized to obtain the adjoint and inverse of the matrices.

Before learning about the details of the transpose of a matrix let’s first learn about “What is a matrix?”. A matrix is nothing but the representation of the set of data in the rectangular array format. In a matrix, data is arranged in specific rows and columns. Various types of matrices exist in Mathematics and are presented in the order of rows × columns. Let’s take an example of the matrix of order 3 × 2 (say A).

A = [Tex]\begin{bmatrix}1 & 2\\ 3 & 4\\ 5 & 6\end{bmatrix} [/Tex]

In this article, we will learn about the transpose of a matrix, its types, properties, symbols, and order, how to find the transpose of a matrix, and examples of it.

What is a Matrix?

A rectangular array of numbers, symbols, or characters assigned to a particular row and column is called a Matrix. The numbers, symbols, or characters present in the matrix are called elements of the matrix. The number of rows and columns present in a matrix determines the order of the matrix. For example if a matrix ‘A’ contains ‘i’ rows and ‘j’ columns then the matrix is represented by [A]i⨯j. Here, i⨯j determines the order of the matrix. Let us see an example of a matrix.

[Tex]\begin{bmatrix}1 & 2\\ 3 & 4\\ 5 & 6\end{bmatrix}_{3\times2} [/Tex]

In the above example, there are three rows and two columns, hence, the order of the matrix is 3⨯2.

Types of Matrices

There are various types of matrices based on the number of rows and columns they have and also due to the specific characteristics shown by them. Let’s see a few of them

• Row Matrix: A Matrix in which there is only one row and no column is called a Row Matrix.
• Column Matrix: A Matrix in which there is only one column and now row is called a Column Matrix.
• Horizontal Matrix: A Matrix in which the number of rows is less than the number of columns is called a Horizontal Matrix.
• Vertical Matrix: A Matrix in which the number of columns is less than the number of rows is called a Vertical Matrix.
• Rectangular Matrix: A Matrix in which the number of rows and columns are unequal is called a Rectangular Matrix.
• Square Matrix: A matrix in which the number of rows and columns are the same is called a Square Matrix.
• Diagonal Matrix: A square matrix in which the non-diagonal elements are zero is called a Diagonal Matrix.
• Zero Matrix: A matrix whose all elements are zero is called a Zero Matrix.
• Unit Matrix: A diagonal matrix whose all diagonal elements are 1 is called a Unit Matrix.
• Symmetric matrix: A square matrix is said to be symmetric if the transpose of the original matrix is equal to its original matrix. i.e. (A^T) = A. 
• Skew-symmetric: A skew-symmetric (or antisymmetric or antimetric[1]) matrix is a square matrix whose transpose equals its negative.i.e. (A^T) = -A. 

What is Transpose of a Matrix?

Transpose of a matrix is a matrix that is obtained by swapping the rows and columns of the given matrix or vice versa, i.e., for the given matrix the elements in rows are interchanged with the elements in columns. For any given matrix A its transpose is denoted as A^t, or A^T.

Transpose of a Matrix Defnition

The Transpose of a matrix is a mathematical operation that involves flipping the rows and columns of the original matrix.

Representation of Transpose of Matrix

A = [a_(ij)]_{m × n}
A^t = [a_(ji)]_{n × m}

here i, j present the position of a matrix element, row- and column-wise, respectively, such that,1 ≤ i ≤ m and 1 ≤ j ≤ n.

Example: For any given matrix A of order 2 × 3 its transpose is?

[Tex] A = \begin{bmatrix} 2 & 5 & 3\\ 4 & 7 & 0 \end{bmatrix} [/Tex]

Solution:

Transpose of A

A^t = [Tex]\begin{bmatrix} 2 & 4 \\ 5 & 7 \\ 3 & 0 \end{bmatrix} [/Tex]

Order of A^t is 3 × 2

Symbol of Transpose Matrix | Transpose Notation

Transpose of a matrix is the operation that flips the matrix over its principal diagonal and interchanges its rows with columns. Transpose of a matrix A is denoted by the notation A’ or A^T or A^t.

Order of Transpose Matrix

The order of a matrix tells the total elements that a matrix contains. It also represents the number of rows and columns in a matrix. Horizontal values represent the rows of the matrix, and vertical values represent the columns of the matrix. For any matrix A_m×n, the order is m×n, i.e., it has m rows and n columns. Therefore, the transpose of matrix A is A^t and its order is n×m, i.e., it has n rows and m columns.

How to Find the Transpose of a Matrix?

Transpose of any matrix can easily be found by changing the values in the rows with the values in the columns. Let’s take an example to understand this in detail.

For any matrix A_2×3, the order is 2×3 which means it has 2 rows and 3 columns.

A =  [Tex] \begin{bmatrix} a & b & c\\ x & y & z \end{bmatrix} [/Tex]

The transpose of matrix A is A^t of the order 3×2 having 3 rows and 2 columns. In the transpose matrix elements of the first row of the given matrix are changed with the first column of the transpose matrix. Similarly, the elements of the second row of the given matrix A are swapped with the second column of the new matrix A^t and so on till the whole matrix is swapped.

A^t = [Tex]\begin{bmatrix} a & x \\ b & y \\ c & z \end{bmatrix} [/Tex]

Transpose of Row and Column Matrix

A matrix that has a single row is known as a row matrix, whereas a matrix that has a single column is known as a column matrix. The transpose of a row matrix is a column matrix and vice versa. For example, if P is a column matrix of order “4 × 1,” then its transpose is a row matrix of order “1 × 4.” If Q is a row matrix of order “1 × 3,” then its transpose is a column matrix of order “3 × 1.”

[Tex]P = \left[\begin{array}{cccc} a & b & c & d\end{array}\right]⇒ P^{t} = \left[\begin{array}{c} a\\ b\\ c\\ d \end{array}\right] [/Tex]

[Tex]Q = \left[\begin{array}{c} p\\ q\\ r \end{array}\right]⇒ Q^{t} = \left[\begin{array}{ccc} p & q & r\end{array}\right] [/Tex]

Transpose of Horizontal and Vertical Matrices

If the number of rows in a matrix is less than the number of columns, then the matrix is known as a horizontal matrix, and if the number of columns in a matrix is less than the number of rows, then the matrix is known as a vertical matrix. The transpose of a horizontal matrix is a vertical matrix and vice versa. For example, if M is a horizontal matrix of order “2 × 3,” then its transpose is a vertical matrix of order “3 × 2.”

[Tex]M = \left[\begin{array}{ccc} 2 & 0 & -1\\ 0 & 3 & 4 \end{array}\right]_{2\times3}⇒ M^{t} = \left[\begin{array}{cc} 2 & 0\\ 0 & 3\\ -1 & 4 \end{array}\right]_{3\times2} [/Tex]

[Tex]N = \left[\begin{array}{ccc} 2 & 3 & 4\\ 4 & 6 & 8\\ 6 & 9 & 12\\ 8 & 12 & 16 \end{array}\right]_{4\times3}⇒ N^{t} = \left[\begin{array}{cccc} 2 & 4 & 6 & 8\\ 3 & 6 & 9 & 12\\ 4 & 8 & 12 & 16 \end{array}\right]_{3\times4} [/Tex]

Transpose of a Symmetric Matrix

A symmetric matrix is like a special kind of pattern where the numbers are arranged in a way that mirrors each other across the diagonal line from the top left to the bottom right. The transpose of a matrix means flipping the matrix over this diagonal line.

For example,

[Tex]\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \\ \end{bmatrix}[/Tex]

The numbers on either side of the diagonal line are the same: 2 is across from 2, 3 is across from 3, and so on. Now, if we take the transpose of this matrix, we simply flip it over the diagonal line. So, the numbers that were originally in rows become columns and vice versa.

[Tex]\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \\ \end{bmatrix}^T = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \\ \end{bmatrix}[/Tex]

Here, the original matrix and its transpose are exactly the same. That’s because when you transpose a symmetric matrix, you get the same matrix back! This is a special property of symmetric matrices.

Transpose of a Diagonal Matrix

A diagonal matrix is like a pattern where the numbers only appear along the diagonal line from the top left to the bottom right, while all other entries are zeros. The transpose of a matrix means flipping the matrix over this diagonal line.

For example,

[Tex]\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \\ \end{bmatrix}[/Tex]

Here, the numbers 2, 3, and 5 appear along the diagonal, while all other entries are zeros. Since a diagonal matrix is already symmetric over its diagonal, the transpose of a diagonal matrix is simply itself:

[Tex]\begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \\ \end{bmatrix}^T = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 5 \\ \end{bmatrix}[/Tex]

Transpose of a Transposed Matrix

When you transpose a matrix, you essentially flip it over its diagonal line. So, transposing a matrix that has already been transposed means flipping it back to its original orientation.

For example,

[Tex]\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{bmatrix} = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \\ \end{bmatrix}[/Tex]

Now, if we take the transpose of this transposed matrix:

[Tex]\left( \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \\ \end{bmatrix} \right)^T = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ \end{bmatrix}[/Tex]

Transpose of a Square Matrix

Square matrices are matrices that have an equal number of rows and columns. for any square matrix A_n×n, its transpose has the same order i.e., the transpose of A, A^t has order n × n. The rows and columns are interchanged in the transpose of a square matrix.

Transpose of a 2 × 2 Matrix

For any 2 × 2 matrices A, 

A = [Tex]\begin{bmatrix} a & x \\ b & y \end{bmatrix} [/Tex]

its transpose is A^t,

A^t =[Tex] \begin{bmatrix} a & b \\ x & y \end{bmatrix} [/Tex]

Example: Find the transpose of matrix A = [Tex]\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} [/Tex]

Solution:

Transpose of the matrix A =[Tex] \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}     [/Tex] is

A^t =  [Tex]\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} [/Tex]

Transpose of a 3 × 3 Matrix

For any 3 × 3 matrices A, 

A = [Tex]\begin{bmatrix} a & x & p \\ b & y & q \\ c & z & r \end{bmatrix} [/Tex]

its transpose is A^t,

A^t =[Tex] \begin{bmatrix} a & b & c \\ x & y & z \\ p & q & r \end{bmatrix} [/Tex]

Example: Find the transpose of matrix A = [Tex]\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} [/Tex]

Solution:

Transpose of the matrix A =[Tex]\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}     [/Tex] is

A^t = [Tex]\begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix} [/Tex]

Determinant of Transpose of a Matrix

The determinant of the transpose of a matrix A is equal to the determinant of A itself, i.e., for any square matrix A

|A| = |A^T|

Transpose of a Matrix Properties

Let’s learn about the important properties of the transpose of a matrix:

• A square matrix “A” of order “n × n” is said to be an orthogonal matrix, if AA^T = A^TA = I, where “I” is an identity matrix of order “n × n.”
• A square matrix “A” of order “n × n” is said to be a symmetric matrix if its transpose is the same as the original matrix, i.e., A^T = A.
• A square matrix “A” of order “n × n” is said to be a skew-symmetric matrix if its transpose is equal to the negative of the original matrix, i.e., A^T = –A.
• Double Transpose of a Matrix: Transpose of the transpose matrix is the original matrix itself.

(A^t)^t = A

• Transpose of Product of Matrices: This property says that 

(AB)^t = B^tA^t

Proof:

If matrices A and B are of orders m × n and n × p, respectively.

and 

A^t and B^t are the transpose of matrices A and B of orders n × m and p × n respectively (from the product rule of matrices).

It implies, if A = [a(ij)], and A^t = [c(ji)]

Then, [c(ji)] = [a(ij)]

and,

If B = [b(jk)], and B^t = [d(kj)]

Then, [d(kj)] = [b(jk)]

Now, from the product rule of matrices, we can write,

AB is m × p matrix and (AB)^t is p × m matrix.

Also, B^t is a p × n matrix, and A^t is an n × m matrix.

This implies that,

(B^t)(A^t) is a p × m matrix.

Therefore,

(AB)^t and (B^t)(A^t) are both p × m matrices.

Now we can write,

(k, i)^th element of (AB)^t = (i, k)^th element of AB 

[Tex]\sum_{j=1}^{n}  a_{ij}  b_{jk}      [/Tex][Tex]\sum_{j=1}^{n}  c_{ji}  d_{kj} [/Tex]

[Tex]\sum_{j=1}^{n}  d_{kj}  c_{ji} [/Tex]

(k, i)th element of (B^t)(A^t)

Therefore, 

the elements of (AB)^t and (B^t)(A^t) are equal.

Therefore,

(AB)^t = (B^t)(A^t)

• Multiplication by Constant: If a matrix is multiplied by a scalar value and its transpose is taken, then the resultant matrix will be equal to the transpose of the original matrix multiplied by the scalar value, i.e., (kA)^t = kA^t, where k is a scalar value.

Proof:

Let us consider a matrix A = [a_ij]_{m × n} and a scalar k.

The order of the given matrix A is m × n.

If matrix A is multiplied by the scalar value k, then all the elements of the matrix are multiplied with this scalar constant k, however, the order of matrix kA remain same, i.e., m × n.

Now, the order of the transpose of the matrix kA, i.e., (kA)^t will be n × m.

As the order of the matrix A is m × n, the order of its transpose matrix, i.e., A^t will be n × m.

If matrix A^t is multiplied by the scalar value k, then the order of the matrix kA^t will also be n × m.

So, the order of the matrices (kA)^t and kA^t is the same, i.e., n × m.

Now, let us prove that the corresponding elements of (kA)^t and kA^t are equal.

The (i, j)th element of (kA)^t will be equal to the (j, i)th element of kA.

(i, j)^th element of (kA)^t = (j, i)^th element of kA

⇒ (i, j)^th element of (kA)^t = (i, j)^th element of kA^t

So, we say that the corresponding elements of (kA)^t and kA^t are equal.

As the order and corresponding elements of (kA)^t and kA^t are equal, 

Therefore, we can conclude that (kA)^t = kA^t.

• Transpose of Addition of Matrices: This property says that. 

(A + B)^t = A^t + B^t

Proof:

Here A and B are two matrices of order m × n

Let, A = [a(ij)] and B = [b(ij)] of order m × n.

So, (A + B) is also of order m × n matrix

Also, A^t and B^t are of order n × m matrices.

So, the Transpose of matrix (A + B) or (A + B)^t is an n × m matrix.

Now we can say, A^t + B^t is also an n × m matrix.

Now, from the transpose rule,
(j, i)th element of (A + B)^t = (i, j)th element of (A + B)

= (i, j)th element of A + (i, j)th element of B
= (j, i)th element of A^t + (j, i)th element of B^t
= (j, i)th element of (A^t + B^t)

Therefore,

(A + B)^t = A^t + B^t

• If “A” is a square matrix of any order and is invertible, then the inverse of its transpose is equal to the transpose of the inverse of the original matrix, i.e., (A^t)^-1 = (A^-1)^t.

Proof:

To prove that (A^t)^-1 = (A^-1)^t, let us consider a non-singular square matrix A.

RHS = (A^-1)^t

Now, multiply (A^-1)^t by A^t

= (A^-1)^t × A^t

We know that (AB)^t = B^tA^t

So, (A^-1)^tA^t = (AA^-1)^t

We know that the AA^-1 = I, where “I” is an identity matrix.

So, (A^-1)^tA^t = I^t

⇒ (A^-1)^tA^t = I (Since, I^t = I)

⇒ (A^-1)^t = (A^t)^-1 = LHS

Hence proved.

Therefore, (A^t)^-1 = (A^-1)^t

People Also Read:

• Adjoint of a Matrix
• Determinant of a Matrix
• Inverse of a Matrix

Solved Examples on Transpose of a Matrix

Example 1: Find the transpose of the matrix A = [Tex]\begin{bmatrix} a & b & c\\ p & q & r \end{bmatrix} [/Tex]

Solution:

The transpose of matrix A is A^t

A^t = [Tex]\begin{bmatrix} a & p \\ b & q \\ c & r \end{bmatrix} [/Tex]

Example 2: For matrices, [Tex]A =  \begin{bmatrix} -2 & 1 & 3\\ 0 & 4 & -1 \end{bmatrix}      [/Tex] and [Tex] B = \begin{bmatrix} 2 & 1 \\-3 & 0 \\ 4 & -5 \end{bmatrix} [/Tex]

Prove that for these matrices hold the property, (AB)^t = (B^t)(A^t)

Solution:

Here A and B are 2 × 3 and 3 × 2 matrices respectively. So, by the product rule of a matrix, we can find their product and the final matrices would be of 2 × 2 matrix.

L.H.S

Now,

[Tex]AB=  \begin{bmatrix} -2 & 1 & 3\\ 0 & 4 & -1 \end{bmatrix} \times  \begin{bmatrix} 2 & 1 \\-3 & 0 \\ 4 & -5 \end{bmatrix}    [/Tex]

[Tex]AB =\begin{bmatrix} (-2)×2+1×(-3)+3×4 & (-2)×1+1×0+3×(-5) \\ 0×2+4×(-3)+(-1)×4 & 0×1+4×0+(-1)×(-5) \end{bmatrix} [/Tex]

[Tex]AB= \begin{bmatrix} 5 & -17 \\ -16 & 5 \end{bmatrix} [/Tex]

So, Transpose of matrix AB is,

[Tex](AB)^{t} =  \begin{bmatrix} 5 & -16 \\ -17 & 5 \end{bmatrix}      [/Tex][Tex]\begin{bmatrix} 5 & -16 \\ -17 & 5 \end{bmatrix} [/Tex]

R.H.S

[Tex]A^{t} =  \begin{bmatrix} -2 & 0 \\ 1 & 4 \\ 3 & -1 \end{bmatrix} [/Tex]

and

[Tex]B^{t} = \begin{bmatrix} 2 & -3 & 4 \\ 1 & 0 & -5 \end{bmatrix} [/Tex]

So,

[Tex]B^{t}A^{t} =  \begin{bmatrix} 2 & -3 & 4\\ 1 & 0 & -5 \end{bmatrix} \times  \begin{bmatrix} -2 & 0 \\ 1 & 4 \\ 3 & -1 \end{bmatrix}\\ B^{t}A^{t} =  \begin{bmatrix} 2×(-2)+(-3)×1+4×3 & 2×0+(-3)×4+4×(-1) \\ 1×(-2)+0×1+(-5)×3 & 1×0+0×4+(-5)×(-1) \end{bmatrix}\\ B^{t}A^{t} =  \begin{bmatrix} 5 & -16 \\ -17 & 5 \end{bmatrix} [/Tex]

Therefore,

(AB)^t = B^tA^t

Example 3: Verify whether (Q^T)^T = Q or not.

[Tex]Q = \left[\begin{array}{cc} 1 & 5\\ 2 & 6\\ 3 & 8 \end{array}\right] [/Tex]

Solution:

[Tex]Q = \left[\begin{array}{cc} 1 & 5\\ 2 & 6\\ 3 & 8 \end{array}\right] [/Tex]

[Tex]Q^{T} = \left[\begin{array}{cc} 1 & 5\\ 2 & 6\\ 3 & 8 \end{array}\right]^{T} = \left[\begin{array}{ccc} 1 & 2 & 3\\ 5 & 6 & 8 \end{array}\right] [/Tex]

[Tex](Q^{T})^{T} = \left[\begin{array}{ccc} 1 & 2 & 3\\ 5 & 6 & 8 \end{array}\right]^{T} [/Tex]

[Tex](Q^{T})^{T} = \left[\begin{array}{cc} 1 & 5\\ 2 & 6\\ 3 & 8 \end{array}\right] = Q [/Tex]

Hence verified.

Example 4: Verify whether the matrix given below is symmetric or not.

[Tex]P = \left[\begin{array}{cc} 6 & -5\\ -5 & 6 \end{array}\right] [/Tex]

Solution:

We know that a square matrix “P” of order “n × n” is said to be a symmetric matrix if its transpose is the same as the original matrix, i.e., P^T = P.

[Tex]P^{T} = \left[\begin{array}{cc} 6 & -5\\ -5 & 6 \end{array}\right]^{T} [/Tex]

Now, P^T is obtained by interchanging its rows into columns.

[Tex]P^{T} = \left[\begin{array}{cc} 6 & -5\\ -5 & 6 \end{array}\right] = P [/Tex]

As P^T = P, the given square matrix is symmetric.

Example 5: For matrices [Tex]A= \begin{bmatrix} -1 & 5 \\ 3 & 2 \end{bmatrix}      [/Tex] and [Tex]B= \begin{bmatrix} 3 & -2 \\5 & 4 \end{bmatrix} [/Tex]

Prove that these matrices hold this property, (A + B)^t = A^t + B^t

Solution:

L.H.S

[Tex](A+B)= \begin{bmatrix} -1 & 5 \\ 3 & 2 \end{bmatrix} + \begin{bmatrix} 3 & -2 \\5 & 4 \end{bmatrix}      [/Tex][Tex]= \begin{bmatrix} (-1)+3 & 5+(-2) \\ 3+5 & 2+4 \end{bmatrix}      [/Tex][Tex]= \begin{bmatrix} 2 & 3 \\ 8 & 6 \end{bmatrix} [/Tex]

So, 

[Tex](A+B)^{t} =  \begin{bmatrix} 2 & 8 \\ 3 & 6 \end{bmatrix} [/Tex]

R.H.S

[Tex]A^{t} =  \begin{bmatrix} -1 & 3 \\ 5 & 2 \end{bmatrix}      [/Tex][Tex] \begin{bmatrix} -1 & 3 \\ 5 & 2 \end{bmatrix} [/Tex]

 and, 

[Tex]B^{t} =  \begin{bmatrix} 3 & 5 \\ -2 & 4 \end{bmatrix} [/Tex]

Now, 

[Tex]A^{t} + B^{t} =  \begin{bmatrix} -1 & 3 \\ 5 & 2 \end{bmatrix} +  \begin{bmatrix} 3 & 5 \\ -2 & 4 \end{bmatrix}      [/Tex][Tex]= \begin{bmatrix} (-1)+3 & 3+5 \\ 5+(-2) & 2+4 \end{bmatrix}      [/Tex][Tex]A^{t} + B^{t} =  \begin{bmatrix} 2 & 8 \\ 3 & 6 \end{bmatrix} [/Tex]

Therefore, 

(A + B)^t = A^t + B^t

FAQs on Transpose of a Matrix

What is the Transpose of a Matrix?

Transpose of a matrix is a matrix that is obtained by interchanging the rows and columns of the matrix. The transpose of matrix A is denoted as A^t. For a given matrix of order m×n, transpose of matrix is of order n×m.

What is the order of the Transpose of a Square Matrix?

For a square matrix order of matrix does not change on transpoe, therefore for a matrix of order n×n, the order of its transpose is also n×n. 

What is the Addition Property of the Transpose Matrix?

Addition property of transpose of matrix states that the sum of two transpose matrices is always equal to the sum of the transpose of individual matrices, i.e.,

(A+B)′ = A′+B′

What is the Multiplication Property of the Transpose Matrix?

Multiplication property of transpose of matrix states that the Product of the transpose of two matrices is always equal to the product of the transpose of individual matrices in reverse order, i.e.,

(A×B)′ = B′ × A′

How to calculate the Transpose of a Matrix?

Transpose of any matrix can easily be found by changing the values in the rows with the values in the columns.