In higher mathematics, a concept such as a transposed matrix is studied. It should be noted: for many it seems that this is a rather complicated topic that cannot be mastered. However, it is not. In order to understand exactly how such an easy operation is carried out, you only need to get a little familiar with the basic concept - the matrix. Any student can understand the topic if he takes the time to study it.
What is a matrix?
Matrices in math are pretty common. It should be noted that they are also found in computer science. Thanks to them and with their help it is easy to program and create software.
What is a matrix? This is the table in which the items are placed. It necessarily has a rectangular appearance. Speaking in the simplest language, the matrix is a table of numbers. It is denoted using any capital Latin letters. It can be rectangular or square. There are also separate rows and columns, which are called vectors. Such matrices get only one line of numbers. In order to understand what size the table has, you need to pay attention to the number of rows and columns. The first is denoted by the letter m, and the second by n.
It must be understood what the diagonal of the matrix is. There are side and main. The second is that strip of numbers that goes from left to right from the first to the last element. In this case, the side will be a line from right to left.
With matrices, you can do almost all the simplest arithmetic operations, that is, add, subtract, multiply among themselves and separately by a number. They can also be transposed.
Transpose process
A transposed matrix is a matrix in which rows and columns are swapped. This is done as easily as possible. It is designated as A with superscript T ( AT T ). In principle, it should be said that in higher mathematics this is one of the simplest operations on matrices. The table size is saved. Such a matrix is called transposed.
Transpose Matrix Properties
In order to correctly make the transposition process, it is necessary to understand what properties of this operation exist.
- There must be an original matrix to any transposed table. Their identifiers must be equal to each other.
- If there is a scalar unit, then when performing this operation it can be removed.
- With double transposition of the matrix, it will be equal to the original.
- If we compare two folded tables with the changed columns and rows, with the sum of the elements over which this operation was performed, they will be the same.
- The last property is that if the tables multiplied among themselves are transposed, then the value should be equal to the results obtained during the multiplication of the transposed matrices among themselves in the reverse order.
Why transpose?
A matrix in mathematics is necessary in order to solve certain problems with it. In some of them, you need to calculate the inverse table. To do this, find the identifier. Next, the elements of the future matrix are calculated, then they are transposed. It remains to find only the directly inverse table. We can say that in such problems it is required to find X, and this can be done quite easily with basic knowledge of the theory of equations.
Summary
This article examined what a transposed matrix is. This topic will be useful to future engineers who need to be able to correctly calculate complex structures. Sometimes the matrix is not so easy to decide, you have to smash your head. However, in the course of student mathematics, this operation is carried out as easily as possible and without any effort.