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Matrix Transpose

By Jazzb | June 25th 2012 | Views:

Let A= [aij] be a matrix of order m x n .Then matrix B=[bij] of order n x m is called the transponse of a matrix A if bij = aij .In words , we can say that (I , j) th element of B is equal to (I , j)th element of A.Alternatively , we can say that a matrix obtained by interchanging the rows and columns of matrix is called the transpose of the matrix .The transpose of matrix A is denoted by A’ or AT.For Example if A = rectangular array [ 1 2// 5 6 //0 4]3x2 , then A’ = rectangular array[1 5 0 // 2 6 4]2 x3 .

Properties of transpose of matrix: Here are four Transpose Matrix properties listed below

1) Transpose of the transpose of a matrix is the matrix itself. For Example , Let A = rectangular array [ 2 3 4// 5 6 7]2 x3 , then A’ = rectangular array [ 2 5 // 3 6 // 4 7]3x2 . and( A’)’ = rectangular array [ 2 3 4// 5 6 7 ]2x3 = A . Then we have (A’)’= A

2) If A is any matrix and k is a scalar , then (kA)’=kA’. For example , if A = rectangular array [1 2 3// 4 5 6] and k = 2 then kA = rectangular array 2[1 2 3// 4 5 6] = rectangular array [2 4 6 //8 10 12]=> (kA)’ = rectangular array[2 8 // 4 10 //6 12] now A’ = rectangular array [1 4 // 2 5//3 6], therefore kA’= 2[1 4 // 2 5 //3 6] = [2 8// 4 10 //6 12], From (i) and (ii) we get (kA)’=kA’

3) For any two matrices A and B of the same order , (A+B )’ = A’ + B’.For example , if A = rectangular array[1 4 0 // 2 3 5] and B = rectangular array [ 3 2 1// 4 -1 3], then A’ = rectangular array [ 1 2// 4 3//0 5] and B’ = rectangular array[ 3 4// 2 -1//1 3]and A+B = rectangular array[ 1 4 0// 2 3 5] + rectangular array [ 3 2 1 //4 -1 3] = rectangular array[4 6 1//6 2 8 ] , therefore (A+B)’ = rectangular array [ 4 6// 6 2//1 8] …..(i) ,also A’ +B’ = rectangular array[ 1 2// 4 3//0 5]+ rectangular array[3 4//2 -1//1 3]= rectangular array [4 6// 6 2 // 1 8]….(ii), from (i) and (ii), we get (A+B)’= A’+B’

4) For any two matrices A and B , (AB)’=B’A’ where product Ab is well defined.For example if A = rectangular array[ 1 2 3// 2 1 3] andB= rectangular array[0 2// 4 3// 1 5], Then AB = rectangular array[ 1 2 3// 2 1 3] * rectangular array[0 2// 4 3// 1 5]= rectangular array[11 23// 7 22], therefore (AB)’= rectangular array[11 7 // 23 22]………(iii) , Now A’= rectangular array[1 2 //2 1 //3 3 ] and B’ = rectangular array [0 4 1 //2 3 5] , therefore B’A’ = rectangular array[0 4 1 // 2 3 5]* rectangular array[1 2// 2 1 // 3 3 ] = rectangular array[11 7 // 23 22] ….(iv) , from (iii)and (iv) we get (AB)’=B’A’

Know more about the Definition of Irrational Numbers,Independent and Dependent Variables Definition,Adjoint of Matrix. Online tutoring will help us to learn and do our homework very easily without going here and there.

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