Transpose of a Matrix

The transpose of a matrix is one of the most common methods used for matrix transformation in matrix concepts across linear algebra. The transpose of a matrix is obtained by changing the rows into columns and columns into rows for a given matrix. It is especially useful in applications where inverse and adjoint of matrices are to be obtained.

In this article, let's learn about the transpose of a matrix, its definition, properties along with solved examples.

The transpose of a matrix is obtained by changing its rows into columns (or equivalently, its columns into rows). A rectangular array of numbers or functions that are arranged in the form of rows and columns is called a matrix. This array of numbers are called either entries or elements of a matrix.

Here for matrix A the elements of the first row have been written in the first column of the new matrix, and the elements of the second row have been written in the second column of the new matrix. And this new matrix is denoted as A^T, which is the transpose of the given matrix A.

Transpose of a Matrix Symbol

In linear algebra, the transpose of a matrix is actually an operator that flips a matrix over its diagonal by switching the row and column indices of matrix B and producing another matrix. Transpose of a matrix B is often denoted by either B' or B^T. Sometimes, they are also denoted as B^tr or B^t. If a matrix B is of order m×n, then the transpose of the matrix B’ is of the order n×m.

The order of a matrix represents the number of rows and columns in a given matrix. The horizontal lines of the elements are all called the rows of the matrix which is denoted by n, and the vertical lines of the elements are called the columns of the matrix which is denoted by m. Together, they represent the order of a matrix, which is written as n × m. And the order of the transpose of the given matrix is written as m x n.

Let us check the below example to understand more clearly about how to find the transpose of a matrix.

A = \(\left[\begin{array}{ll}
-2 & 5 & 6 \\
5 & 2 & 7
\end{array}\right]\)

In the above example, we can see that the given matrix of order 2 × 3. The elements of the first row [-2, 5, 6] are written in the first column, and the elements in the second row [5, 2, 7] are written in the second column to obtain the transpose matrix. The transpose of the matrix A is A^T and has an order of 3 x 2.

A^T = \(\left[\begin{array}{ll}
-2 & 5 \\
5 & 2 \\6&7
\end{array}\right]\)

The matrix that is resulting from a given matrix B after changing or reversing its rows to columns and columns to rows is called the transpose of a matrix B. Let us look at the transpose of 2 × 2 and 3 × 3 square matrices.

Transpose of a 2 × 2 Matrix

Let us consider a 2 × 2 matrix C, after interchanging the rows and columns, the resultant transpose of the matrix C^T looks like:

Same way, we can find the transpose of a matrix A as:

\(A=\left[\begin{array}{ll}
1 & -2 \\
3 & -4
\end{array}\right]\)

After interchanging the rows and columns, the resultant transpose of the matrix A^T looks like:

A^T = \(\left[\begin{array}{ll}
1 & 3 \\
-2 & -4
\end{array}\right]\)

Transpose of a 3 × 3 Matrix

Let us consider a 3 × 3 matrix C:

\(C = \left[\begin{array}{ccc}
c_{11} & c_{12} & c_{13} \\
c_{21} & c_{22} & c_{23} \\
c_{31} & c_{32} & c_{33}
\end{array}\right] \)

After reversing the rows and columns, the resultant transpose of the matrix C^T looks like:

C^T = \(\left[\begin{array}{ccc}
c_{11} & c_{21} & c_{31} \\
c_{12} & c_{22} & c_{32} \\
c_{13} & c_{23} & c_{33}
\end{array}\right] \)

Same way, we can find the transpose of a matrix A as:

\(A = \left[\begin{array}{ccc}
1 & 2 & -3 \\
4 & -5 & 6 \\
7 & 8 & -9
\end{array}\right] \)

After reversing the rows and columns, the resultant transpose of the matrix A^T looks like:

A^T = \(\left[\begin{array}{ccc}
1 & 4 & 7 \\
2 & -5 & 8 \\
-3 & 6 & -9
\end{array}\right] \)

Transpose of a matrix is used in some of the linear transformations as they reveal some of the important properties of the transformation. Let's learn about some of the important properties of the transpose of a matrix:

• A square matrix B of order n × n is considered to be an orthogonal matrix, only when B^T × B = B × B^T = I, here I is an identity matrix.
• A square matrix B of order n × n is considered to be a symmetric matrix if its transpose is equal to itself. i.e., B^T = B.
• A square matrix B of order n × n is considered to be a skew-symmetric matrix, only when its transpose is equal to its negative. i.e., B^T = -B.
• The transpose of the sum/difference (B ± C)^T is the sum/difference of transposes of the matrices B and C. i.e., B^T ± C^T = (B ± C)^T.
• The transpose of an invertible matrix B is also invertible, and its inverse is actually the transpose of the inverse of the original matrix B. This can be represented as: (B^T)^-1 = (B^-1)^T.
• Transpose when applied to a matrix, has higher precedence than multiplication and addition operations i.e.,
  CB^T = C(B^T) and
  C + D^T = C + (D^T)

Addition Property of the Transpose of a Matrix

Consider two matrices B and C, the transpose of the sum (B + C)^T is the sum of transposes of the matrices B and C. This can be represented as (B + C)^T = B^T+C^T. Let's consider an example here:

B = \(\left[\begin{array}{ll}
2 & 3 \\
1 & 0
\end{array}\right]\)

C = \(\left[\begin{array}{ll}
4 & -2 \\
3 & 5
\end{array}\right]\)

B + C = \(\left[\begin{array}{ll}
(2+4) & (3-2) \\
(1+3) & (0+5)
\end{array}\right]\) = \(\left[\begin{array}{ll}
6 & 1 \\
4 & 5
\end{array}\right]\)

(B+C)^T = \(\left[\begin{array}{ll}
6 & 4 \\
1 & 5
\end{array}\right]\)

Now let's take the transpose of the matrices separately,

B^T = \(\left[\begin{array}{ll}
2 & 1 \\
3 & 0
\end{array}\right]\)

C^T = \(\left[\begin{array}{ll}
4 & 3 \\
-2 & 5
\end{array}\right]\)

B^T+ C^T =\(\left[\begin{array}{ll}
(2+4) & (1+3) \\
(3-2) & (0+5)
\end{array}\right]\) = \(\left[\begin{array}{ll}
6 & 4 \\
1 & 5
\end{array}\right]\)

B^T+ C^T = \(\left[\begin{array}{ll}
6 & 4 \\
1 & 5
\end{array}\right]\)

We can see from the above example that the sum remains the same in both cases. Thus, transpose operation respects addition.

A matrix is considered to be horizontal when the number of rows in the matrix is less than the number of columns in that matrix. And, a matrix is considered to be vertical, when the number of columns in the matrix is less than the number of rows in that matrix. Let's consider a horizontal matrix P and a vertical matrix Q as:

P = \(\left[\begin{array}{ll}
1 & 2 & 3 \\
4 & 5 & 6
\end{array}\right]\)

P^T = \(\left[\begin{array}{ccc}
1 & 4 \\
2 & 5 \\
3 & 6
\end{array}\right] \)

Q = \(\left[\begin{array}{ccc}
1 & 4 \\
2 & 5 \\
3 & 6
\end{array}\right] \)

Q^T = \(\left[\begin{array}{ll}
1 & 2 & 3 \\
4 & 5 & 6
\end{array}\right]\)

From the above two examples, we can see that the transpose of a horizontal matrix P results in a vertical matrix P^T and the transpose of a vertical matrix Q results in a horizontal matrix Q^T.

A square matrix of order n × n is considered to be symmetric, if and only if it is symmetric about its diagonal. A square matrix C of size n x n is considered to be symmetric if and only if C^T = C. Consider the two given symmetric matrices A and B:

A = \(\left[\begin{array}{ll}
2 & -1 \\
-1 & 2
\end{array}\right]\)

A^T = \(\left[\begin{array}{ll}
2 & -1 \\
-1 & 2
\end{array}\right]\)

B = \(\left[\begin{array}{ccc}
2 & 3 & 6 \\
3 & 4 & 5 \\
6 & 5 & 9
\end{array}\right] \)

B^T = \(\left[\begin{array}{ccc}
2 & 3 & 6 \\
3 & 4 & 5 \\
6 & 5 & 9
\end{array}\right] \)

We can see from the above example that, after taking the transposes of the two matrices A and B, they are equal to their original matrices i.e., A = A^T and B = B^T.

A square matrix of order n × n is considered to be a diagonal matrix, if and only if it is all the elements except the diagonal elements are zero. Consider the two given diagonal matrices C and D:

C = \(\left[\begin{array}{ll}
5 & 0 \\
0 & -5
\end{array}\right]\)

C^T = \(\left[\begin{array}{ll}
5 & 0 \\
0 & -5
\end{array}\right]\)

D = \(\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
\end{array}\right] \)

D^T = \(\left[\begin{array}{ccc}
1 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
\end{array}\right] \)

We can see from the above two examples that the two diagonal matrices C and D remain as diagonal matrices even after the transpose is applied.

The transpose of a transposed matrix is itself. i.e., for any matrix B, (B^T)^T = B. Let's consider two examples here:

C = \(\left[\begin{array}{ll}
1 & -2 \\
3 & -4
\end{array}\right]\)

C^T = \(\left[\begin{array}{ll}
1 & 3 \\
-2 & -4
\end{array}\right]\)

(C^T)^T = \(\left[\begin{array}{ll}
1 & -2 \\
3 & -4
\end{array}\right]\)

D = \(\left[\begin{array}{ccc}
1 & 2 & -3 \\
4 & -5 & 6 \\
7 & 8 & -9
\end{array}\right] \)

D^T = \(\left[\begin{array}{ccc}
1 & 4 & 7 \\
2 & -5 & 8 \\
-3 & 6 & -9
\end{array}\right] \)

(D^T)^T = \(\left[\begin{array}{ccc}
1 & 2 & -3 \\
4 & -5 & 6 \\
7 & 8 & -9
\end{array}\right] \)

From the above two examples, we can see that the transpose of the already transposed matrix, yielded the original matrix.

The determinant of the transpose of a matrix A is equal to the determinant of A itself. i.e., det A = det A^T, for any square matrix A. For more information, you can click here.

The adjoint of a square matrix B is the transpose of the cofactor matrix C of the original B. The relationship between the adjoint of the original matrix B and the transpose of a cofactor matrix C can be represented as adj(B) = (C)^T. Let's consider this example:

Consider the 2×2 matrix D:

D = \(\left[\begin{array}{ll}
3 & 6 \\
-4 & 8
\end{array}\right]\)

The minor matrix M can be shown as:

\(M=\left[\begin{array}{ll}
8 & -4 \\
6 & 3
\end{array}\right]\)

The co-factor matrix C can be shown as:

\(C=\left[\begin{array}{ll}
8 & 4 \\
-6 & 3
\end{array}\right]\)

The transpose C^T of the co-factor matrix can be shown as:

adj(D) = C^T = \(\left[\begin{array}{ll}
8 & -6 \\
4 & 3
\end{array}\right]\)

Important Notes on Transpose of 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 matrix B and producing another matrix.
• Transpose of a matrix B is often denoted by either B' or B^T. Sometimes, they are also denoted as B^tr or B^t.
• If a matrix B is of order m x n, then the transpose of the matrix B’ is of the order n x m.

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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 B is denoted by B^T.

How To Find the Transpose of a Matrix?

The transpose of any given matrix can be calculated by interchanging its rows and columns. Consider a 2 × 2 matrix B:

\(A=\left[\begin{array}{ll}
1 & -2 \\
3 & 7
\end{array}\right]\)

After interchanging the rows and columns, the resultant transpose of the matrix C^T looks like:

A^T = \(\left[\begin{array}{ll}
1 & 3 \\
-2 & 7
\end{array}\right]\)

What Is the Addition Property of the Transpose of a Matrix?

As per the addition property of the transpose of a matrix, for the two matrices B and C, the transpose of the sum (B + C)^T is the sum of transposes of the matrices B and C. This can be represented as (B + C)^T = B^T+C^T.

What Is the Multiplication Property of the Transpose of a Matrix?

As per the multiplication property of the transpose of a matrix, the transpose when applied to a matrix has the higher precedence than multiplication and addition operations i.e., CB^T = C(B^T) and C + D^T = C + (D^T)

What Are the Different Properties of the Transpose of Matrix?

Here are the different properties of the transpose of a matrix:

• A square matrix B of order n × n is considered to be an orthogonal matrix, only when B × B^T = I, here I is an identity matrix.
• A square matrix B of order n × n is considered to be a skew-symmetric matrix, only when its transpose B^T = -B i.e., it is equal to its negative.
• The transpose of the difference (B - C)^T is the difference of transposes of the matrices B and C. B^T - C^T = (B - C)^T
• The transpose of an invertible matrix B is also invertible, and its inverse B^-1 is actually the transpose of the inverse of the original matrix B. This can be represented as: (B^T)^-1 = (B^-1)^T.
• Transpose when applied to a matrix, has the higher precedence than multiplication and addition operations i.e., CB^T = C(B^T) and C + D^T = C + (D^T)

What Is the Determinant of the Transpose of a Matrix?

The determinant of the transpose of a square matrix of order n×n is equal to the determinant of the matrix i.e., |B^T| = |B|.

What Is the Transpose of a Square Matrix?

For any square matrix of order n×n, this is how the transpose is applied to the matrix. Consider a 2 × 2 matrix C:

\(C=\left[\begin{array}{ll}
c_{11} & c_{12} \\
c_{21} & c_{22}
\end{array}\right]\)

After interchanging the rows and columns, the resultant transpose of the matrix C^T looks like:

C^T = \(\left[\begin{array}{ll}
c_{11} & c_{21} \\
c_{12} & c_{22}
\end{array}\right]\)

How To Find the Inverse of a Matrix by the Transpose Method?

Here are the steps to be followed to calculate the inverse of a matrix D by using the transpose method:

• Find the determinant of |D|. If |D| = 0, then the inverse does not exist. Only if |D| ≠ 0, the inverse exists.
• Find the minor matrix M of all the elements of the matrix D
• Find the cofactor matrix C of all the minor elements of the matrix M
• Find the adj D by taking the transpose of the cofactor matrix C
• Then, find the inverse of the matrix D as D^-1 = (1/|D|) × adj(D)
• Check if the inverse is correct by verifying it as D × D^-1 = I, where I is an identity matrix.