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Transpose of a Matrix

\( \) \( \) \( \)

Definition of the Transpose of a Matrix

If \( M \) is an \( m \times n \) matrix, then the transpose of \( M \), denoted by \( M^T \), is the \( n \times m \) matrix obtained by interchanging the rows and columns of matrix \( M \).

These are examples of the transpose of matrices.

a)

\(
A =
\begin{bmatrix}
1 & -2 & -3\\
\end{bmatrix}
\)

Matrix \( A \) has one row and a size (or order) \( 1 \times 3 \). The transpose of matrix \( A \) is obtained by interchanging the row of the matrix into a column. Hence the transpose of matrix \( A \) has a size \( 3 \times 1 \) and denoted by \( A^T \) is given by

\(
A^T =
\begin{bmatrix}
1\\
-2\\
-3
\end{bmatrix}
\)

b)

\(
B =
\begin{bmatrix}
9\\
-2\\
6 \\

\end{bmatrix}
\)

The transpose of matrix \( B \), which has one column and a size \( 4 \times 1 \), is obtained by interchanging the column of the matrix into a row. Hence the transpose of matrix \( B \) has an order \( 1 \times 4 \) and denoted by \( B^T \) is given by

\(
B^T =
\begin{bmatrix}
9&
-2&
6&

\end{bmatrix}
\)

c)

\(
C =
\begin{bmatrix}
-3 & 0 & – 2\\
6 & 4 & 0
\end{bmatrix}
\)

Matrix \( C \) has a size \( 2 \times 3 \). The transpose of this matrix is obtained by interchanging the rows of the matrix into columns (or columns into rows). Hence the transpose \( C^T \) of matrix \( C \) has an order \( 3 \times 2 \) and is given by

\(
C^T =
\begin{bmatrix}
-3 & 6 \\
0 & 4 \\
– 2 & 0
\end{bmatrix}
\)

d)

\(
D =
\begin{bmatrix}
2 & – 5 & 9 \\
-7 & 0 & 9 \\
1 & -2 & 11
\end{bmatrix}
\)

The transpose of matrix \( D \) with order \( 3 \times 3 \) is obtained by interchanging the rows of the matrix into columns (or columns into rows). Hence the transpose \( D^T \) of matrix \( D \) has an order \( 3 \times 3 \) and is given by

\(
D^T =
\begin{bmatrix}
2 & -7 & 1 \\
– 5 & 0 & -2 \\
9 & 9 & 11
\end{bmatrix}
\)

Note that the rows the transpose of a given matrix are the columns of the matrix and the columns of the transpose are the rows of the matrix.

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Properties of the Transpose of Matrices

Some of the most important properties of the transpose of matrices are given below.

  1.   \( (A^T)^T = A \).
  2.   \( (AB)^T = B^T A^T \)
  3.   \( (A+B)^T = A^T + B^T \)
  4.   \( (k A)^T = k A^T \) , k is a real number.
  5.   \( (A^T)^{-1} = (A^{-1})^T \)
  6.   \( Det(A^T) = Det(A) \).
  7.   \( A^T = A \) if and only if \( A \) is a symmetric matrix.
  8.   \( A^{-1} = A^T \) if and only if \( A \) is an orthogonal (square) matrix.

Examples with Solutions

Example 1

Find the transpose of the matrices:

a) \( A =
\begin{bmatrix}
0 & 0 & 1 \\
1 & 0 & 0
\end{bmatrix} \)      b) \( B =
\begin{bmatrix}
-2 & 0 & 1 \\
\end{bmatrix} \)      c) \( C =
\begin{bmatrix}
5 & -7 \\
1 & -4 \\
0 & -1 \\
7 & -4 \\
\end{bmatrix} \)

Solution

We find the transpose of a matrix by interchanging the rows and columns as follows:

a) \( A^T =
\begin{bmatrix}
0 & 1\\
0 & 0 \\
1 & 0
\end{bmatrix} \)      b) \( B^T =
\begin{bmatrix}
-2 \\
0 \\
1
\end{bmatrix} \)      c) \( C^T =
\begin{bmatrix}
5 & 1 & 0 & 7 \\
-7 & -4 & -1 & -4
\end{bmatrix} \)

Example 2

Matrices \( A \) and \( B \) are given by \( A =
\begin{bmatrix}
-1 & 0 & 1 \\
1 & 2 & 0
\end{bmatrix} \) , \( B =
\begin{bmatrix}
-1 \\
1 \\
2
\end{bmatrix} \).

Show that \( (AB)^T = B^T A^T \) (verify property 2 above).

Solution

Calculate \( AB \)

\( AB =
\begin{bmatrix}
-1 & 0 & 1 \\
1 & 2 & 0
\end{bmatrix}
\begin{bmatrix}
-1 \\
1 \\
2
\end{bmatrix}
=
\begin{bmatrix}
3 \\
1
\end{bmatrix}
\)

Determine \( (AB)^T \)

\( (AB)^T = \begin{bmatrix}
3 &
1
\end{bmatrix} \)

Determine \( A^T \) and \( B^T \)

\( A^T =
\begin{bmatrix}
-1 & 1 \\
0 & 2 \\
1 & 0
\end{bmatrix}
\) , \( B^T = \begin{bmatrix}
-1 & 1 & 2
\end{bmatrix} \)

Calculate \( B^T A^T \)

\( B^T A^T =
\begin{bmatrix}
-1 & 1 & 2
\end{bmatrix}
\begin{bmatrix}
-1 & 1 \\
0 & 2 \\
1 & 0
\end{bmatrix}
=
\begin{bmatrix}
3 &
1
\end{bmatrix} \)

Hence \( (AB)^T = B^T A^T \)

Example 3

Let matrix \( A =
\begin{bmatrix}
-1 & 0\\
1 & 2
\end{bmatrix} \)

Show that \( (A^T)^{-1} = (A^{-1})^T \) (verify property 5 above).

Solution

Determine \( A^T \)

\( A^T =
\begin{bmatrix}
-1 & 1\\
0 & 2
\end{bmatrix} \)

Use the formula of the inverse of a \( 2 \times 2 \) matrix \( \begin{bmatrix}
x & y \\
z & w \\
\end{bmatrix}^{-1} = \dfrac{1}{xw – yz}
\begin{bmatrix}
w & – y \\
– z & x \\
\end{bmatrix} \) to find \( (A^T)^{-1} \)

\( (A^T)^{-1} = -\dfrac{1}{2} \begin{bmatrix}
2 & 0\\
1 & -1
\end{bmatrix} \)

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\( \quad \quad =
\begin{bmatrix}
-1 & 0\\
-\dfrac{1}{2} & \dfrac{1}{2}
\end{bmatrix} \)

Use the same formula of the inverse of a \( 2 \times 2 \) matrix given above to find \( A^{-1} \)

\( A^{-1} = -\dfrac{1}{2}
\begin{bmatrix}
2 & 1\\
0 & -1
\end{bmatrix}
\)

\( \quad \quad = \begin{bmatrix}
-1 & -\dfrac{1}{2}\\
0 & \dfrac{1}{2}
\end{bmatrix} \)

We now determine \( (A^{-1}) ^T \)

\( (A^{-1}) ^T =
\begin{bmatrix}
-1 & 0 \\
-\dfrac{1}{2} & \dfrac{1}{2}
\end{bmatrix} \)

Hence we conclude that \( (A^T)^{-1} = (A^{-1})^T \).

Example 4

Let matrix \( A =
\begin{bmatrix}
-1 & 0 & 1\\
1 & 2 & 2 \\
-2 & 0 & 1
\end{bmatrix} \)

Show that \( Det(A^T) = Det(A) \) (verify property 6 above)

Solution

Calculate \( Det(A) \) using the upper row

Calculate \( Det(A) = -1 ( 2 \times 1 – 0) + 1 (0 – 2(-2) ) = 2 \)

Determine \( A^T \)

\( A^T
\begin{bmatrix}
-1 & 1 & -2 \\
0 & 2 & 0 \\
1 & 2 & 1
\end{bmatrix}
\)

Calculate \( Det(A^T) \) using the leftmost column

\( Det(A^T) = -1 ( 2 \times 1 – 0) + 1 ( 0 – 2(-2)) = 2
\)

Hence we conclude that \( Det(A^T) = Det(A) \)

Example 5

Use property 7 above to show that matrix \( A =
\begin{bmatrix}
0 & -1 & 3\\
-1 & 2 & 5 \\
3 & -5 & 1
\end{bmatrix} \)
is not a symmetric matrix.

Solution

Determine \( A^T \)

\( A^T =
\begin{bmatrix}
0 & -1 & 3 \\
-1 & 2 & -5 \\
3 & 5 & 1
\end{bmatrix}
\)

We can verify that the entry at \( A_{2,3} = 5 \) and the entry at \( A^T_{2,3} = – 5 \), therefore matrices \( A^T \) and \( A \) are not equal which means that matrix \( A \) is not symmetric.

Questions (with solutions given below)

  • Part 1

    Given matrix
    \( A = \begin{bmatrix}
    1 & 1 & -3 \\
    -1 & 0 & 0
    \end{bmatrix} \), calculate and show that the matrices \( A^T A \) and \( A A^T \) are both symmetric.

  • Part 2

    Given the matrices
    \( A = \begin{bmatrix}
    -5 & -2 \\
    – 1 & 2
    \end{bmatrix} \) and
    \( B = \begin{bmatrix}
    0 & 3 \\
    2 & -3
    \end{bmatrix} \) , find the matrix \( (A+B)^T \)

  • Part 3

    Let matrix \( A = \begin{bmatrix}
    0 & \dfrac{5}{3 \sqrt 5} & \dfrac{2}{3} \\
    \dfrac{1}{\sqrt 5} & – \dfrac{4}{3 \sqrt 5} & \dfrac{2}{3} \\
    \dfrac{2}{\sqrt 5} & \dfrac{2}{3 \sqrt 5} & – \dfrac{1}{3}
    \end{bmatrix} \). Show that \( A A^T = A^T A = \begin{bmatrix}
    1 & 0 & 0\\
    0 & 1 & 0 \\
    0 & 0 & 1
    \end{bmatrix} \) and therefore \( A^{-1} = A^T \).


More References and links


Transpose of Matrix | Logical Programming in C | by Mr.Srinivas


Transpose of Matrix | Logical Programming in C | by Mr.Srinivas
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Transpose of Matrix | Logical Programming in C | by Mr.Srinivas

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How to find Adjoint of 3 X  3 Matrix

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Transpose matrix | Lecture 4 | Matrix Algebra for Engineers


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The Excel TRANSPOSE Function


Learn how to transpose columns into rows in Excel using the power of the TRANSPOSE function. In other words, you’ll learn how to make your column headings become row headings, and vice versa. There’s more than one way to accomplish this, but in this video we’ll focus on using a formula to transpose. My entire playlist of Excel tutorials: http://bit.ly/tech4excel Consider supporting Technology for Teachers and Students on Patreon: https://www.patreon.com/technologyforteachersandstudents Here’s the practice file that goes with this video: https://bit.ly/transposeexcel exceltutorial

The Excel TRANSPOSE Function

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ขอบคุณที่รับชมกระทู้ครับ transpose of a matrix

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