CHAPTER 04.01: UNARY MATRIX OPERATIONS: Transpose of a Matrix
In this segment we will talk about what do we mean by a transpose of a matrix. So let A be a m by n matrix. Then B is called transpose of A if bji is same as aij. So what you basically are doing is that you are exchanging the row and the column. So whatever is the ith row, jth element in A becomes the jth row ith element in B. So you can very well see also then the size of the matrix will be n by m; so the size of the matrix B will be n rows and m columns. And weíll see that through an example as well. Also, what people say is that hey Ė if we can write B as the transpose of A, we can write B is equal to A transpose. So thatís another was of symbolizing this.
So letís take an example of finding a transpose. So letís suppose we take 5, 6, 7; 9, 19, 29; 31, 34, 39; 61, 67, 73. So letís suppose somebody says hey Ė this is youíre a matrix. Now what you want to do is you want to find the transpose of this A matrix. So the first thing which you have to realize is that this has four rows and three columns. So 1-2-3-4. So four rows and three columns. So a transpose of the matrix will be now four rows and three columns. No, three rows and hour columns. So weíll have three rows and four columns because we had four rows and three columns in A, and now we have three rows and four columns in the transpose of the matrix.
So this is the first row first column, so this stays as the first row first column because weíre going to switch row number and the column number of a11 is still 11. However, if you go to this particular element which is first row 2nd column, this becomes second row first column of the transpose, which would be six right here. And the same thing here Ė this one is the first row third column, but in the transpose it will become the third row first column. SO that would be 7 right here. So you can see very well that this will become 9, 19, 29, 31, 34, 39, and then 61, 67, 73. So thatís how the transpose of this particular matrix is going to look like.
Letís do a spot check here. So letís suppose I have this element as 39. This is the third row, third column, and 39 is right here. Thatís also third row, third column there. Letís look at another element right here. This is fourth row, second column, and see if this 67 is the second row, fourth column of the transpose, which it is here. Thatís where 67 is and that place is second row, fourth column. So youíre finding out thatís the transpose of the matrix. So all of these elements have ben where the row number and column number have been interchanged so far as elements are concerned. And that is the end of this segment.