## What is the Transpose of these Matrices ??

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As it is clear from its name  transpose means trans + pose i. e. transfer + position ( transfer of position ) ,  the transpose   of any matrix is obtained by transfer of Rows into Columns And vice versa. i.e transforming 1st row to 1st column and transforming 2nd row to 2nd column and so on for 3rd ,4th and 5th rows and columns. This is very easy and interesting topic in matrices and determinants.How to find the Transpose of Matrix ,

## To find the transpose of the matrix

1st of all  shift all the elements which are in 1st row to 1st column as
5
5
2
,then shift the elements which are in      2nd row to 2nd column as

-1
-3
7
similarly  shift all the elements which are in 3rd row to 3rd column as
4
2
8

And the matrix so obtained is the transpose matrix. We can  check that same colour row have been  transformed to same colour column

Now we shall take one  example to find the transpose matrix
Here Given matrix have 3 rows and 4 columns .It means we shall have 4 rows and 3  columns  in transpose matrix.
1st shift all the elements which are in 1st row to 1st column as
5
8
6
-4
shift all the elements which are in 2nd row to 2nd column as
3
5
8
3
shift all the elements which are in 3rd row to 3rd column as
-3
8
-7
6
so shifting the corresponding Rows into Corresponding columns. We can  check that same colour rows are transformed to same colour columns.

Now we shall take one more  example to find the transpose of matrix .Here Given matrix have 3 rows and 4 columns,It means  we shall have 4 rows and  3 columns in transpose of that matrix.

1st shift all the elements which are in 1st row to 1st column as
7
-1
-2
5
shift all the elements which are in 2nd row to 2nd column as
4
4
3
3
shift all the elements which are in 3rd row to 3rd column as
8
9
6
-1
so shifting the corresponding Rows into Corresponding columns. We can that check same colour rows are transformed to same colour column.

Let us take an example  Where  A =

Step 1 Then  on transforming 1st Column  to 1st Row  ,we have

4             -3         9          as 1st  Row

Step 2 Then  on transforming  2nd Column  to 2nd Row  ,we have

5             2          -2          as  2nd Row

||ly    on transforming   3rd  Column  to 3rd Row  ,we have

7          3          8             as 3rd Row

After  taking Transpose  A' will be

Let us take more example to find out the transpose of matrices. These are two examples , both of which are of  3×3 orders. Hence the transpose of these matrices will be  again 3×3.

1st consider matrix G , after transforming its 1st row into column , the 1st column of the transpose matrix of  G '  will be  -2,  -5,  4.
After transforming its 2nd row into column , the 2nd column of the transpose matrix of  G '  will be   -5,  7 ,  3. And after transforming its 3rd row into column , the 3rd  column of the transpose matrix of  G '  will be  4,  3,  8. If we write the transpose of  matrix G then we can see that there is no difference between the matrix G and the transpose of the matrix G.

Now consider matrix H , after transforming its 1st row into column , the 1st column of the transpose matrix of  G '  will be  2,  3,  4.
After transforming its 2nd row into column , the 2nd column of the transpose matrix of  H '  will be   3, 5 ,  6. And after transforming its 3rd row into column , the 3rd  column of the transpose matrix of  H '  will be  4,  6,  7. Again in the case of matrix H , Matrix H and its Transpose matrix   H '  are same.

Here is more interesting Example of Transpose of this Matrix,This matrix have 3 Rows and 3 columns,after taking Transpose this matrix still have 3 Rows and 3 columns,
Now Take Transpose of this Matrix
what ?
Surprise to see that the Transpose of some of the Matrices are the Matrices Itself, i. e. if A¹= A
Such Matrices are called Symmetric  Matrices.

Again if we take the transpose of the matrix given below and take -1 common from the matrix so obtained i .e  this matrix will be equal to transpose of the negative of the transpose of the given matrix
then   A¹ =  - A ,  Such Matrices are called  Skew Matrices.

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