Matrix Transposition

A very useful operation in matrix algebra is that of transposition. The transpose of a matrix A, is the matrix in which the rows of the original matrix A become columns and the columns of A become rows. The transpose is denoted by Ar.

Definition 8.10

The transpose matrix, AT, is the original matrix A with its rows and columns interchanged.

This implies that A' will have its dimensions reversed when compared with A.

Example 8.14 Find the transpose of the 2x3 matrix 4, given by

Solution

Since A is 2 x 3. its transpose will be 3 x 2. Placing the first row of A. (I 2 3). as the first column of Ar and the second row of -4. (2 5 7|, as the second column of Ar yields i 2

Definition 8.11

A matrix A that is equal to its transpose A' is called a symmetric matrix.

An example of a symmetric matrix is

where transposing rows and columns shows that A' =

Since equality of matrices, as we have seen earlier, implies that all the elements of the two matrices in each position have to be equal, then the orders of A and A7 have to be the same. In that case, A has to be a square matrix and so. of course, is A1.

From now on. we will define all vectors as column vectors, taking their transposes when we want to have a row vector.

Example 8.15 The Profit Function

Using all the information of example 8.9. and expressing all the vectors as column vectors, we obtain the profit function as n=p7q-wri ■

Properties of Transposes

Below we present some useful properties of the transpose matrix by a series of theorems and examples.

Theorem 8.2

The transpose of the transpose matrix (A1 ) ' is the original mauix A-

Solution

Theorem 8.3 The transpose of a sum of matrices is the sum of the transposes:

Example 8.17 Compute (A + B)r, for A and B given below:

Solution

We first compute Ar and Br, and then A + B and (A -+- B)r.

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