Бесплатный фрагмент - Cool Matrices. Issue No. 4

Preface:

This problem book is designed for people who are concerned about their intellectual health. After all, modern life requires that our minds be constantly in shape. But how to maintain your mind in a fairly effective state, that is, how to maintain your memory and your mind, that is, «thinking»? There are many such ways: solving scanwords, crosswords, solving puzzles. Scanword puzzles and crossword puzzles help boost your memory, help develop your horizons, and puzzles train your mind well. I suggest solving mathematical matrices, this is a very good way to train your memory and train your mind. Below is an example of solving mathematical matrices. The rules for solving them will also be outlined, discussed with an example. Of course, you can, if you know how to solve matrices, solve them in your own way, but I still recommend solving matrices according to two rules:

1) Try to do all actions in your mind, write only intermediate results.

2) Solve matrices, only one per day.

If you try to do all the actions in your mind, then you will have to multiply and add in your mind, so the method of solving matrices is given in this form, this is what strengthens a person’s mind.

If you solve matrices constantly and in large quantities, you will quickly get tired of it, and you will quit this activity. But you don’t need this, you need to keep your mind in good shape, so solve one or two a day maximum, you can also solve matrices before complex intellectual work, but no more than one or two. The matrices are the simplest 3 by 3, so if you find it difficult to decide in your head, don’t despair, everything will come with time. These puzzles are suitable for all ages. Keep your mind sharp. Good luck to you. Mathematics is very cool!

Example solution:

Find the determinant of the matrix:

Solution:

Let’s reduce the matrix to triangular form:

Definition: A matrix is triangular if all matrix values below or above the main diagonal are zero.

Definition: The main diagonal of a matrix is the elements whose row index matches the column index.

Let us reduce this matrix D to triangular form:

This means that the elements are: a21=0, a31=0, a32=0. Or elements: a12=0, a13=0, a23=0.That is, elements below and above the main diagonal.

Or

Let us reduce matrix D to triangular form, that is, we reduce it using elementary transformations:

Before we multiply each row of the matrix by a number, so that all elements of the first column are equal to the same number. To do this, take the elements of the first column: 5, 2, 8. And in each factor we remove the number that is contained in the first column against each line:

Then we get:

Now multiply the first line by (-1) in such a way as to get elements in the first line that will be negative: