Transpose Of Matrix - Meaning, Properties, Examples - Cuemath

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.

1.	What is the Transpose of a Matrix?
2.	Order of Transpose Matrix
3.	Transpose of a Square Matrix
4.	Properties of Transpose of a Matrix
5.	Transpose of a Horizontal and a Vertical Matrix
6.	Transpose of a Symmetric Matrix
7.	Transpose of a Diagonal Matrix
8.	Transpose of a Transposed Matrix
9.	Determinant of Transpose of a Matrix
10.	Relation Between Adjoint and Transpose Matrix
11.	FAQs on Transpose of Matrix

What is the Transpose of a Matrix?

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 AT, 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 BT. Sometimes, they are also denoted as Btr or Bt. If a matrix B is of order m×n, then the transpose of the matrix B’ is of the order n×m.

Order of Transpose Matrix

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 AT and has an order of 3 x 2.

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

Transpose of a Square Matrix

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 CT 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 AT looks like:

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

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 CT looks like:

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

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 AT looks like:

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

Properties of Transpose of a Matrix

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 BT × B = B × BT = 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., BT = 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., BT = -B.
• The transpose of the sum/difference (B ± C)T is the sum/difference of transposes of the matrices B and C. i.e., BT ± CT = (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: (BT)-1 = (B-1)T.
• Transpose when applied to a matrix, has higher precedence than multiplication and addition operations i.e., CBT = C(BT) and C + DT = C + (DT)

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 = BT+CT. 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,

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

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

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

BT+ CT = \(\left[\begin{array}{ll} 6 & 4 \\ 1 & 5 \end{array}\right]\)

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

Transpose of a Horizontal and a Vertical Matrix

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

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

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

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

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

Transpose of a Symmetric Matrix

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 CT = C. Consider the two given symmetric matrices A and B:

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

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

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

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

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 = AT and B = BT.

Transpose of a Diagonal Matrix

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

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

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

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

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.

Transpose of a Transposed Matrix

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

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

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

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

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

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

(DT)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.

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., det A = det AT, for any square matrix A. For more information, you can click here.

Relation Between Adjoint and Transpose Matrix

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 CT of the co-factor matrix can be shown as:

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

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 BT. Sometimes, they are also denoted as Btr or Bt.
• 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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