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# Transpose of a Matrix

What if two matrices have the same set of elements? Will they be necessarily considered equal? Certainly not! This is because their order may be different. Transpose is a matrix formed by swapping the rows into columns and vice-versa. Sounds interesting right? Let’s learn about it in further more detail.

### Suggested Videos

Introduction to Matrices
Inverse of a Matrix
Special Matrices

## Definition

The new matrix obtained by interchanging the rows and columns of the original matrix is called as the transpose of the matrix. If A = [aij] be an m × n matrix, then the matrix obtained by interchanging the rows and columns of A would be the transpose of A. of It is denoted by A′or (AT). In other words, if A = [aij]mxn ,thenA′ = [aji]nxm . For example,

$$A =\begin{bmatrix} 3 & -5 \\ 4 & 7/2 \\ 9 & 5/8 \end{bmatrix} A’ =\begin{bmatrix} 3 & 4 & 9 \\ -5 & 7/2 & 5/8\end{bmatrix}$$

## Properties

### 1) Transpose of Transpose of a Matrix

The transpose of the transpose of a matrix is the matrix itself: (AT)T = A. For example,

Verify that (AT)T = A. It is determined as shown below:

Therefore,

### 2) Transpose of a Scalar Multiple

The transpose of a matrix times a scalar (k) is equal to the constant times the transpose of the matrix: (kA)T = kAFor example, $$Let \: A =\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$ and k = 2. Verify that (kA)T = kAT

Solution: LHS = (kA)T

$$= (2 ×\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix})^T$$

$$=( \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}) ^ T$$

$$=\begin{bmatrix} 2 & 6 \\ 4 & 8 \end{bmatrix}$$

RHS = kAT

$$= 2 ×\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^T$$

$$= 2 ×\begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$$

$$=\begin{bmatrix} 2 & 6 \\ 4 & 8 \end{bmatrix}$$

Therefore LHS = RHS. Hence, (kA)T = kAT

### 3) Transpose of a Sum

The transpose of the sum of two matrices is equivalent to the sum of their transposes: (A + B)T = AT + BT. For example:

,

verify that (A ± B)T = AT ± BT.

Therefore,

The transpose matrices for A and B are given as below:

Therefore,

Hence (A ± B)T = AT ± BT.

### 4) Transpose of a Product

The transpose of the product of two matrices is equivalent to the product of their transposes in reversed order: (AB)T = BT AT . The same is true for the product of multiple matrices: (ABC)T = CTBTAT. For example,

,

verify that (AB)T = BT AT.

Solution: The product of A and B is:

Therefore,

If we take the transpose of A and B separately and multiply A with B, then we have:

Hence (AB)T = BT AT .

Learn the Types of Matrix here.

## Solved Examples For You

Question 1: If matrix A is a circulant matrix whose elements of first row are a, b, c all > 0 such that abc = 1 and ATA = I, then a3 + b3+ c3 equals

1. 0
2. 3
3. 1
4. 4

Answer :Given, A is a circulant matrix of elements a, b, c and abc = 1. So, $$A =\begin{bmatrix} a & b & c\\ c & a & b\\ b & c & a \end{bmatrix}$$

So, det A = a(a2 – bc) – b(ac-b2) + c(c2 – ab) = a3 + b3 + c3 – 3 and ATA = I
|ATA| = |I| = 1
|A| |AT| = 1
|A|2 = 1
Therefore, |A| = ±1. So, det A = ± 1. After substituting the value in the det A = a3 + b3 + c3 – 3abc, we get, a3 + b3 + c3 = 4 or  a3 + b3 + c3 = -2. Therefore, answer is option D.

Question 2: What is a transpose?

Answer: The new matrix that we attain by interchanging the rows and columns of the original matrix is referred to as the transpose of the matrix.

Question 3: Is transpose and inverse the same?

Answer: A matrix has an inverse if and only if it is both squares as well as non-degenerate. Thus, this inverse is unique. Moreover, the inverse of an orthogonal matrix is referred to as its transpose. They are the only matrices that have inverses as same as their transpositions.

Question 4: Can you transpose a non-square matrix?

Answer: Yes, you can transpose a non-square matrix. However, you just have to make sure that the number of rows in mat2 must match the number of columns in the mat and vice versa. In other words, if the mat is an NxM matrix, then mat2 must come out as an MxN matrix.

Question 5: What is the transpose of a vector?

Answer: The transpose, which we indicate by T, of a row vector, refers to a column vector. Moreover, the transpose of a column vector is a row vector. Further, the set of all row vectors creates a vector space which we refer to as row space. Likewise, the set of all column vectors makes a vector space which we refer to as column space.

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