A matrix in math is an ordered rectangular array of numbers consisting of $m$ horizontal rows and $n$ vertical columns. There are various types of matrices and the transpose of a matrix is one such type.
And these are used in multiple areas such as finding solutions to linear equations, in electronic spreadsheet programs for personal computers, and in physical operations such as magnification, rotation, and reflection through a plane.
Let’s understand what is transpose of a matrix and its properties with examples.
What is Transpose of a Matrix?
A rectangular array of numbers or functions that are arranged in the form of rows and columns is called a matrix. The numbers in the matrix are called either entries or elements of a matrix. The transpose of a matrix is obtained by interchanging its rows and columns (i.e., making rows as columns and columns as rows).
Example of a Transpose of a Matrix
Let’s consider some examples to understand the transpose of a matrix.
Example 1: If $\text{A}$ is a matrix given by $\text{A} = \begin{bmatrix} a & b & c\\ d & e & f \end{bmatrix}$, then transpose of matrix $\text{A}$, is denoted by $\text{A}^{’}$ or $\text{A}^{\text{T}}$ and is written as $\text{A}^{’}(\text{ or }\text{A}^{\text{T}}) = \begin{bmatrix} a & d\\ b & e\\ c & f \end{bmatrix}$.
Example 2: If $\text{B}$ is a matrix given by $\text{B} = \begin{bmatrix} 1 & -2\\ 4 & 7\\ 9 & 0 \end{bmatrix}$, then transpose of matrix $\text{B}$, is denoted by $\text{B}^{’}$ or $\text{B}^{\text{T}}$ and is written as $\text{B}^{’}(\text{ or }\text{B}^{\text{T}}) = \begin{bmatrix} 1 & 4 & 9\\ -2 & 7 & 0 \end{bmatrix}$.
Example 3: If $\text{C}$ is a matrix given by $\text{C} = \begin{bmatrix} 5 & 3 & 7\\ -2 & 1 & 0\\ 4 & -3 & 8 \end{bmatrix}$, then transpose of matrix $\text{C}$, is denoted by $\text{C}^{’}$ or $\text{C}^{\text{T}}$ and is written as $\text{C}^{’}(\text{ or }\text{C}^{\text{T}}) = \begin{bmatrix} 5 & -2 & 4\\ 3 & 1 & -3\\ 7 & 0 & 8 \end{bmatrix}$.
Order of a Transpose of a Matrix
From the above examples, we observe that the rows of a matrix become the columns of the transpose of a matrix and the columns of a matrix become the rows of the transpose of a matrix. The transpose of a matrix is actually an operator that flips a matrix over its diagonal by switching the row and column indices of a matrix and producing another matrix.
Transpose of a matrix $\text{A}$ is often denoted by either $\text{A}^{’}$ or $\text{A}^{\text{T}}$. Sometimes, they are also denoted as $\text{A}^{\text{tr}}$ or $\text{A}^{\text{t}}$. If a matrix $\text{A}$ is of order $m \times n$, then the transpose of the matrix $\text{A}^{’}$ is of the order $n \times m$.
Examples of Order of a Transpose of a Matrix
Example 1: If a matrix $\text{X}$ is given by $[x_{ij}]_{2 \times 3}$, then what is the order of a matrix $\text{X}^{’}$?
Order of matrix $\text{X}$ is $2 \times 3$, therefore the order of matrix $\text{X}^{’}$ is $3 \times 2$.
Example 2: If $\text{P}$ is a vertical matrix, then $\text{P}^{’}$ is a _________ matrix.
a) vertical
b) horizontal
Since $\text{P}$ is a vertical matrix, therefore, $\text{Number of rows } \gt \text{ Number of columns}$. Now when we take the transpose of a matrix $\text{P}$, then $\text{Number of columns } \gt \text{ Number of rows}$ and thus $\text{P}^{’}$ is a horizontal matrix.
Example 3: If $\text{X}$ is a square matrix, then $\text{X}^{’}$ is a _________ matrix.
a) vertical
b) horizontal
c) None of these
The answer is c) None of these.
Since $\text{X}$ is a square matrix, therefore, $\text{Number of rows } = \text{ Number of columns}$. Now when we take the transpose of a matrix $\text{X}$, then also $\text{Number of rows }= \text{ Number of columns}$ and thus $\text{X}^{’}$ is a square matrix.
Transpose of the Transpose Matrix
The transpose of the transposed matrix is itself. i.e., for any matrix $\text{A}$, $(\text{A}^{\text{T}})^{\text{T}} = \text{A}$. If $\text{A}$ is a matrix given by $\text{A} = \begin{bmatrix} a & b & c\\ d & e & f \end{bmatrix} $, then the transpose of matrix $\text{A}$ is a matrix given by $\text{A}^{’} = \begin{bmatrix} a & d\\ b & e \\ c & f \end{bmatrix}$. Now again taking the transpose of the matrix $\text{A}^{’}$, we get $(\text{A}^{’})^{’} = \begin{bmatrix} a & b & c\\ d & e & f \end{bmatrix}$.
Therefore, you can observe that $(\text{A}^{\text{T}})^{\text{T}} = \text{A}$ or $(\text{A}^{‘})^{‘} = \text{A}$.
Addition Property of Transpose Matrix
The transpose of a sum of two matrices of the same dimensions(or order) is equal to the sum of the transpose of two matrices. For two two matrices $\text{A}$ and $\text{B}$, $(\text{A} + \text{B})^{’} = \text{A}^{’} + \text{B}^{’}$.
Examples of Addition Property of Transpose Matrix
Example 1: $\text{A}$ and $\text{B}$ are two matrices such that $\text{A} = \begin{bmatrix} 4 & -1 \\ 2 & 6 \end{bmatrix}$ and $\text{B} = \begin{bmatrix} 1 & 5 \\ -6 & 2 \end{bmatrix}$. Find
(i) $\text{A}^{’}$
(ii) $\text{B}^{’}$
(iii) $(\text{A} + \text{B})^{’}$
(iv) $\text{A}^{’} + \text{B}^{’}$
(v) Verify that $(\text{A} + \text{B})^{’} = \text{A}^{’} + \text{B}^{’}$
$\text{A} = \begin{bmatrix} 4 & -1 \\ 2 & 6 \end{bmatrix}$ and $\text{B} = \begin{bmatrix} 1 & 5 \\ -6 & 2 \end{bmatrix}$.
(i) Transpose of matrix $\text{A}$, i.e., $\text{A}^{’} = \begin{bmatrix} 4 & 2 \\ -1 & 6 \end{bmatrix}$.
(ii) Transpose of matrix $\text{B}$, i.e., $\text{B}^{’} = \begin{bmatrix} 1 & -6 \\ 5 & 2 \end{bmatrix}$.
(iii) $(\text{A} + \text{B})^{’} = \left( \begin{bmatrix} 4 & -1 \\ 2 & 6 \end{bmatrix} + \begin{bmatrix} 1 & 5 \\ -6 & 2 \end{bmatrix} \right)^{‘} = \left(\begin{bmatrix} 4 + 1 & -1 + 5 \\ 2 – 6 & 6 + 2\end{bmatrix} \right)^{‘} = \left(\begin{bmatrix} 5 & 4 \\ -4 & 8\end{bmatrix} \right)^{‘} = \begin{bmatrix} 5 & -4 \\ 4 & 8 \end{bmatrix}$
(iv) $\text{A}^{’} + \text{B}^{’} = \begin{bmatrix} 4 & 2 \\ -1 & 6 \end{bmatrix} + \begin{bmatrix} 1 & -6 \\ 5 & 2 \end{bmatrix} = \begin{bmatrix} 4 + 1 & 2 – 6 \\ -1 + 5 & 6 + 2 \end{bmatrix} = \begin{bmatrix} 5 & -4 \\ 4 & 8 \end{bmatrix}$
(v) From (iii) and (iv), we see that $(\text{A} + \text{B})^{’} = \text{A}^{’} + \text{B}^{’}$.
Multiplication Property of Transpose Matrix by a Scalar
If a matrix is multiplied by a scalar(or constant) and its transpose is taken, then the matrix obtained is equal to the transpose of the original matrix multiplied by that scalar(or constant). For a matrix $\text{A}$, $(k \text{A})^{’}$ = $k \text{A}^{’}$, where $k$ is a scalar.
Examples of Multiplication Property of Transpose Matrix by a Scalar
Example 1: For a matrix $\text{A} = \begin{bmatrix} 3 & 1 \\ 7 & 2 \end{bmatrix} $, find
(i) $5 \text{A}$
(ii) $5 \text{A}^{’}$
(iii) $(5 \text{A})^{’}$
(iv) Verify that $5 \text{A}^{’} = (5 \text{A})^{’}$
$\text{A} = \begin{bmatrix} 3 & 1 \\ 7 & 2 \end{bmatrix} $
(i) $5 \text{A} = 5\begin{bmatrix} 3 & 1 \\ 7 & 2 \end{bmatrix} = \begin{bmatrix} 5 \times 3 & 5 \times 1 \\ 5 \times 7 & 5 \times 2 \end{bmatrix} = \begin{bmatrix} 15 & 5 \\ 35 & 10 \end{bmatrix}$
(ii) $\text{A}^{’} = \begin{bmatrix} 3 & 7 \\ 1 & 2 \end{bmatrix}$
Therefore $5 \text{A} = 5\begin{bmatrix} 3 & 7 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 15 & 35 \\ 5 & 10 \end{bmatrix}$
(iii) Taking transpose of a matrix in (i), we get $\begin{bmatrix} 15 & 35 \\ 5 & 10 \end{bmatrix}$
(iv) From (iii) and (iv), we see that $5 \text{A}^{’} = (5 \text{A})^{’}$
Multiplication Property of Transpose Matrix
The transpose of the product of two matrices is equal to the product of the transpose of the two matrices in reverse order. For two matrices, $\text{A}$
Examples of Multiplication Property of Transpose Matrix
Example 1: For two matrices $\text{A} = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}$ and $\text{B} = \begin{bmatrix} 2 & 3 \\ 2 & 5 \end{bmatrix}$, find
(i) $\text{AB}$
(ii) $\text{A}^{’}$
(iii) $\text{B}^{’}$
(iv) $\text{B}^{’} \text{A}^{’}$
(v) Verify that $(\text{AB})^{‘} = \text{B}^{’} \text{A}^{’}$
$\text{A} = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix}$ and $\text{B} = \begin{bmatrix} 2 & 3 \\ 2 & 5 \end{bmatrix}$
(i) $\text{A} \text{B} = \begin{bmatrix} 1 & 2 \\ 3 & 1 \end{bmatrix} \begin{bmatrix} 2 & 3 \\ 2 & 5 \end{bmatrix} = \begin{bmatrix} 1 \times 2 + 2 \times 2 & 1 \times 3 + 2 \times 5 \\ 3 \times 2 + 1 \times 2 & 3 \times 3 + 1 \times 5 \end{bmatrix} = \begin{bmatrix} 2 + 4 & 3 + 10 \\ 6 + 2 & 9 + 5 \end{bmatrix} = \begin{bmatrix} 6 & 13 \\ 8 & 14 \end{bmatrix}$
(ii) $\text{A}^{’} = \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix}$
(iii) $\text{B}^{’} = \begin{bmatrix} 2 & 2 \\ 3 & 5 \end{bmatrix}$
(iv) $\text{B}^{’} \text{A}^{’} = \begin{bmatrix} 2 & 2 \\ 3 & 5 \end{bmatrix} \begin{bmatrix} 1 & 3 \\ 2 & 1 \end{bmatrix} = \begin{bmatrix} 2 \times 1 + 2 \times 2 & 2 \times 3 + 2 \times 1 \\ 3 \times 1 + 5 \times 2 & 3 \times 3 + 5 \times 1 \end{bmatrix} = \begin{bmatrix} 2 + 4 & 6 + 2 \\ 3 + 10 & 9 + 5 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 13 & 14 \end{bmatrix}$
(v) From (i), we get $(\text{AB})^{‘} = \begin{bmatrix} 6 & 8 \\ 13 & 14 \end{bmatrix}$. Therefore, $(\text{AB})^{‘} = \text{B}^{’} \text{A}^{’}$
Transpose of a Symmetric Matrix
A square matrix of order $n \times n$ is called symmetric, if and only if it is symmetric about its diagonal. The transpose of a symmetric matrix is equal to the symmetric matrix, i.e., if $\text{A}$ is a symmetric matrix, then $\text{A}^{‘} = \text{A}$.
Examples of Transpose of a Symmetric Matrix
Example 1: A symmetric matrix $\text{A}$ is given as $\text{A} = \begin{bmatrix} 1 & -2 \\ -2 & 1 \end{bmatrix}$. Find $\text{A}^{‘}$ and verify that $\text{A} = \text{A}^{‘}$.
$\text{A} = \begin{bmatrix} 1 & -2 \\ -2 & 1 \end{bmatrix}$.
Interchanging the rows and columns, we get $\text{A}^{‘} = \begin{bmatrix} 1 & -2 \\ -2 & 1 \end{bmatrix}$. We see that $\text{A} = \text{A}^{‘}$.
Example 2: A symmetric matrix $\text{B}$ is given as $\text{B} = \begin{bmatrix} 1 & 2 & 5 \\ 0 & -4 & 7 \\ 3 & 1 & -8 \end{bmatrix}$. Find $\text{B}^{‘}$ and verify that $\text{B} = \text{B}^{‘}$.
$\text{B} = \begin{bmatrix} 1 & 2 & 5 \\ 0 & -4 & 7 \\ 3 & 1 & -8 \end{bmatrix}$.
Interchanging the rows and columns, we get $\text{B}^{‘} = \begin{bmatrix} 1 & 2 & 5 \\ 0 & -4 & 7 \\ 3 & 1 & -8 \end{bmatrix}$.
We see that $\text{B} = \text{B}^{‘}$.
Transpose of a Diagonal Matrix
A square matrix of order $n \times n$ is considered to be a diagonal matrix, if and only if it is all the elements except the diagonal elements are zero. The transpose of a diagonal matrix is equal to the diagonal matrix itself, i.e., if $\text{A}$ is a diagonal matrix, then $\text{A}^{‘} = \text{A}$.
Examples of Transpose of a Diagonal Matrix
Example 1: A diagonal matrix $\text{A}$ is given as $\text{A} = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix}$. Find $\text{A}^{‘}$ and verify that $\text{A} = \text{A}^{‘}$.
$\text{A} = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix}$.
Interchanging the rows and columns, we get $\text{A}^{‘} = \begin{bmatrix} 3 & 0 \\ 0 & 5 \end{bmatrix}$. We see that $\text{A} = \text{A}^{‘}$.
Example 2: A diagonal matrix $\text{B}$ is given as $\text{B} = \begin{bmatrix} 7 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 2 \end{bmatrix}$. Find $\text{B}^{‘}$ and verify that $\text{B} = \text{B}^{‘}$.
$\text{B} = \begin{bmatrix} 7 & 0Â & 0 \\ 0 & -4Â & 0 \\ 0 & 0Â & 2 \end{bmatrix}$.
Interchanging the rows and columns, we get $\text{B}^{‘} = \begin{bmatrix} 7 & 0 & 0 \\ 0 & -4 & 0 \\ 0 & 0 & 2 \end{bmatrix}$.
We see that $\text{B} = \text{B}^{‘}$.
Properties of Transpose of a Matrix
The transpose of a matrix is commonly used in some of the linear transformations as they reveal some of the important properties of the transformation. The important properties of the transpose of a matrix are
- A square matrix $\text{A}$ of order $n \times n$ is considered to be an orthogonal matrix, only when $\text{A}^{\text{T}} \times \text{A} = \text{A} \times \text{A}^{\text{T}} = \text{I}$, where $\text{I}$ is an identity matrix.
- A square matrix $\text{A}$ of order $n \times n$ is considered to be a symmetric matrix if its transpose is equal to itself. i.e., $\text{A}^{\text{T}} = \text{A}$.
- A square matrix $\text{A}$ of order $n \times n$ is considered to be a skew-symmetric matrix, only when its transpose is equal to its negative. i.e., $\text{A}^{\text{T}} = – \text{A}$.
- The transpose of the sum/difference $ \left(\text{A} \pm \text{B} \right)^{\text{T}}$ is the sum/difference of transposes of the matrices $\text{A}$ and $\text{B}$, i.e., $\text{A}^{\text{T}} \pm \text{B}^{\text{T}} = \left(\text{A} \pm \text{B} \right)^{\text{T}}$.
- The transpose of an invertible matrix $\text{B}$ is also invertible, and its inverse is actually the transpose of the inverse of the original matrix $\text{B}$. This can be represented as $\left(\text{A}^{\text{T}} \right)^{-1} = \left(\text{A}^{-1} \right)^{\text{T}}$.
- Transpose when applied to a matrix, has higher precedence than multiplication and addition operations i.e.,
- $\text{AB}^{\text{T}} = \text{A}( \text{B}^{\text{T}})$ and
- $\text{A} + \text{B}^{\text{T}} = \text{A} + (\text{B}^{\text{T}})$
Key Takeaways
- The transpose of a matrix is an operator that flips a matrix over its diagonal by switching the row and column indices of matrix $\text{A}$ and producing another matrix.
- Transpose of a matrix $\text{A}$ is often denoted by either $\text{A}^{‘}$ or $\text{A}^{\text{T}}$. Sometimes, it is also denoted as $\text{A}^{\text{tr}}$ or $\text{B}^{\text{t}}$.
- If a matrix $\text{A}$ is of order $m \times n$, then the transpose of the matrix $\text{A}^{‘}$ is of the order $n \times m$.
Practice Problems
For matrices $\text{A} = \begin{bmatrix} 7 & -2 \\ 4 & 5 \end{bmatrix}$ and $\text{B} = \begin{bmatrix} 3 & 6 \\ 1 & 0 \end{bmatrix}$, find
- $\text{A}^{‘}$
- $\text{B}^{‘}$
- $\text{A}^{‘} + \text{B}^{‘}$
- $(\text{AB})^{‘}$
- $(\text{BA})^{‘}$
- $\text{A}^{‘} \text{B}^{‘}$
FAQs
What is the meaning of transpose of a matrix?
The transpose of a matrix is a matrix that is obtained after changing or reversing its rows to columns (or columns to rows). The transpose of $\text{A}$ is denoted by $\text{A}^{\text{T}}$ or $\text{A}^{‘}$.
How to find the transpose of a matrix?
The transpose of any given matrix can be calculated by interchanging its rows and columns. For example, if $\begin{bmatrix} 2 & -3 \\ 5 & 9 \end{bmatrix}$ is a matrix of order $2 \times 2$, then its transpose is $\begin{bmatrix} 2 & 5 \\ -3 & 9 \end{bmatrix}$.
What is the addition property of the transpose of a matrix?
The addition property of the transpose of a matrix states that for the two matrices, $\text{A}$ and $\text{B}$, the transpose of the sum $(\text{A} + \text{B})^{‘}$ is the sum of transposes of the matrices $\text{A}$ and $\text{B}$. This can be represented as $(\text{A} + \text{B})^{‘} = \text{A}^{‘}+ \text{B}^{‘}$.
What is the multiplication property of the transpose of a matrix?
The multiplication property of the transpose of a matrix states that the transpose of the product of two matrices is equal to the product of the transpose of the two matrices in reverse order. For two matrices, $\text{A}$ and $\text{A}$, $(\text{AB})^{‘} = \text{B}^{’} \text{A}^{’}$.
Conclusion
The transpose of a matrix is obtained by interchanging its rows and columns (i.e., making rows as columns and columns as rows). The transpose of a matrix is used in solving linear equations using the matrix method.
