Transpose of a Matrix

A Matrix is a rectangular arrangement of numbers (or elements) in rows and columns. It is often used in mathematics to represent data, solve systems of equations, or perform transformations. A matrix is written as:

A = \begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}

Here, A is a 3 × 3 matrix(3 rows and 3 columns).

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.

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?

 A = \begin{bmatrix} 2 & 5 & 3\\ 4 & 7 & 0 \end{bmatrix}

Solution:

Transpose of A

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

Order of A^t is 3 × 2

Symbol of Transpose Matrix | Transpose Notation

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 represents the number of rows and columns it contains, written as m×n, where m is the number of rows, and n is the number of columns.

For any matrix A_m×n:

• The transpose of A, denoted as A^T, swaps the rows and columns.
• The order of A^T becomes n×m (i.e., A^T has n rows and m columns).

How to Find the Transpose of a Matrix?

The 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 =   \begin{bmatrix} a & b & c\\ x & y & z \end{bmatrix}

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 = \begin{bmatrix} a & x \\ b & y \\ c & z \end{bmatrix}

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."

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]

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

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."

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}

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}

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,

\begin{bmatrix}1 & 2 & 3 \\2 & 4 & 5 \\3 & 5 & 6 \\\end{bmatrix}

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.

\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}

Here, the original matrix and its transpose are 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.

Read More: Symmetric and Skew 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,

\begin{bmatrix}2 & 0 & 0 \\0 & 3 & 0 \\0 & 0 & 5 \\\end{bmatrix}

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:

\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}

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,

\begin{bmatrix}1 & 2 & 3 \\4 & 5 & 6 \\\end{bmatrix}=\begin{bmatrix}1 & 4 \\2 & 5 \\3 & 6 \\\end{bmatrix}

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

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

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 = \begin{bmatrix} a & x \\ b & y \end{bmatrix}

its transpose is A^t,

A^t = \begin{bmatrix} a & b \\ x & y \end{bmatrix}

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

Solution:

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

A^t =  \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}

Transpose of a 3 × 3 Matrix

For any 3 × 3 matrices A, 

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

its transpose is A^t,

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

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

Solution:

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

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

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

• 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.

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

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

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Solved Examples on Transpose of a Matrix

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

Solution:

The transpose of matrix A is A^t 

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

Example 2: Verify whether (Q^T)^T = Q or not.

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

Solution:

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

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]

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

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

Hence verified.

Example 3: Verify whether the matrix given below is symmetric or not.

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

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.

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

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

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

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