Transpose of a Matrix

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Class 10 Additional Maths | Transpose of Matrices

New Millennium Academy

Pokhara-17, Birauta | Class 10 Additional Maths

Topic: Transpose of Matrices Chapter: Matrix Operations Session: Class 10

Transpose of Matrices

This note is designed with New Millennium Academy navy and gold styling for Class 10 Additional Maths, covering activity, observation, definition, and properties of transpose.

Activity

Let us consider a matrix:

$$A = \begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 7 \end{bmatrix}$$

Observation

  • Matrix $A$ has 2 rows and 3 columns.
  • The order of matrix $A$ is $2 \times 3$.
  • It is a rectangular matrix because rows $\neq$ columns.
Thinking: What will happen if we write elements of rows as columns and columns as rows?
$$\begin{bmatrix} 2 & 5 \\ 3 & 6 \\ 4 & 7 \end{bmatrix}$$
Key Observation: A new matrix is formed with reversed order $3 \times 2$.

Conclusion

We call this new matrix the transpose of the given matrix $A$.

Definition

A new matrix formed by interchanging rows and columns of a given matrix is called the transpose of a matrix.

Let $A$ be a matrix of order $m \times n$. Its transpose is written as $A'$ or $A^t$ with order $n \times m$.

Properties of Transpose of Matrices

Quick Access:

Property 1 Property 2 Property 3 Property 4 Property 5 Property 6

Property 1

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

$$\text{Let } A = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 6 \\ 7 & 9 \end{bmatrix}$$
$$A + B = \begin{bmatrix} 3 & 9 \\ 11 & 14 \end{bmatrix}$$
$$(A + B)^t = \begin{bmatrix} 3 & 11 \\ 9 & 14 \end{bmatrix}$$
$$A^t = \begin{bmatrix} 2 & 4 \\ 3 & 5 \end{bmatrix}, \quad B^t = \begin{bmatrix} 1 & 7 \\ 6 & 9 \end{bmatrix}$$
$$A^t + B^t = \begin{bmatrix} 3 & 11 \\ 9 & 14 \end{bmatrix}$$

Therefore, $(A + B)^t = A^t + B^t$.

Property 2

$(A - B)^t = A^t - B^t$

$$\text{Let } A = \begin{bmatrix} 6 & 2 \\ 3 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 5 \\ 0 & 7 \end{bmatrix}$$
$$A - B = \begin{bmatrix} 2 & -3 \\ 3 & -6 \end{bmatrix}$$
$$(A - B)^t = \begin{bmatrix} 2 & 3 \\ -3 & -6 \end{bmatrix}$$
$$A^t = \begin{bmatrix} 6 & 3 \\ 2 & 1 \end{bmatrix}, \quad B^t = \begin{bmatrix} 4 & 0 \\ 5 & 7 \end{bmatrix}$$
$$A^t - B^t = \begin{bmatrix} 2 & 3 \\ -3 & -6 \end{bmatrix}$$

Therefore, $(A - B)^t = A^t - B^t$.

Property 3

$(KA)^t = KA^t$ where $K$ is a scalar.

$$\text{Let } K = 2, \quad A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$
$$KA = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}$$
$$(KA)^t = \begin{bmatrix} 2 & 6 \\ 4 & 8 \end{bmatrix}$$
$$A^t = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$$
$$KA^t = \begin{bmatrix} 2 & 6 \\ 4 & 8 \end{bmatrix}$$

Therefore, $(KA)^t = KA^t$.

Property 4

$(A^t)^t = A$

$$\text{Let } A = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}$$
$$A^t = \begin{bmatrix} 3 & 5 \\ 4 & 6 \end{bmatrix}$$
$$(A^t)^t = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix} = A$$

Property 5

If $A = A^t$, then the matrix is called a symmetric matrix.

$$\text{Let } A = \begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}$$
$$A^t = \begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix} = A$$

Therefore, $A = A^t$.

Note: If a matrix is equal to its transpose then it is called a symmetric matrix.

Property 6

Transpose of Identity Matrix and Null Matrix.

$$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$
$$I^t = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I$$
$$N = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$
$$N^t = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = N$$
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New Millennium Academy, Pokhara-17, Birauta

Class 10 | Additional Mathematics | Transpose of Matrices

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