# Transpose Of A Matrix

If A = [a

Hence if A = [a

_{ij}] be a matrix of order m x n, then the matrix obtained by interchanging the rows and columns of A is known as the transpose of A. It is represented by A^{T}.Hence if A = [a

_{ij}]_{mxn}, then A^{T}= [a_{ji}]_{mxn}**Example:**If A =, then A^{T}=**PROPERTIES OF TRANSPOSE OF THE MATRICES:**

If A and B be two matrices of any suitable order then,

- (A
^{T})^{T}= A - (A ± B)
^{T}= A^{T}± B^{T} - (kA)
^{T}= kA^{T} - (AB)
^{T}= B^{T}A^{T}

**Example:** Find the transpose of the matrix
A = and verify that**Example:**If A = , verify that**(A ± B)T = A**.^{T}± B^{T}

**Example:**If A = , then verify that**(kA)**where k is any constant.^{T}= kA^{T}

**Example:**If and , verify that**(AB)**.^{T}= B^{T}A^{T}

**(A**.

^{T})^{T}= A**Solution:**

By interchanging the rows and columns of the matrix A we get the transpose of matrix A. Hence transpose of

matrix A = A

^{T}= .

Now (A

^{T})

^{T}= = A

Hence,

**(A**

^{T})^{T}= A**Solution:**

Thus,

**(A + B)**

^{T}= A^{T}+ B^{T}**Solution:**

Given that A = , then

Then

Hence

**(kA)**

^{T}= kA^{T}**Solution:**

Hence

B

^{T}A

^{T}=

Hence

**(AB)**

^{T}= B^{T}A^{T}**Online Transpose of Matrix Help:**

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