The matrix is a special subject in mathematics. It is depicted in the form of a rectangular or square table, composed of a certain number of rows and columns. In mathematics, there is a wide variety of types of matrices, varying in size or content. The numbers of its rows and columns are called orders. These objects are used in mathematics to organize the records of systems of linear equations and conveniently search for their results. The equations using the matrix are solved by the method of Karl Gauss, Gabriel Cramer, minor and algebraic complements, as well as in many other ways. The basic skill when working with matrices is to bring to a standard view. However, for starters, let's look at what types of matrices mathematicians emit.
Zero type
All components of this type of matrix are zeros. Meanwhile, the number of rows and columns is completely different.
Square type
The number of columns and rows of this type of matrix is the same. In other words, it is a table of the form "square". The number of its columns (or rows) is called the order. Special cases are the existence of a second-order matrix (2x2 matrix), fourth-order (4x4), tenth (10x10), seventeenth (17x17) and so on.
Column Vector
This is one of the simplest types of matrices, containing only one column, which includes three numerical values. It represents a series of free terms (numbers independent of variables) in systems of linear equations.
String Vector
A view similar to the previous one. It consists of three numerical elements, which in turn are organized on one line.
Diagonal type
The numerical values in the diagonal form of the matrix are accepted only by the components of the main diagonal (highlighted in green). The main diagonal begins with an element located in the upper left corner, and ends with an element in the lower right, respectively. The remaining components are zero. The diagonal type is only a square matrix of any order. Among the matrices of the diagonal form, we can distinguish scalar. All its components take the same values.
Unit matrix
A subspecies of the diagonal matrix. All its numerical values are units. Using a single type of matrix tables, perform its basic transformations or find the matrix inverse to the original.
Canonical type
The canonical form of the matrix is considered one of the main ones; casting to it is often necessary for work. The number of rows and columns in the canonical matrix is different; it does not necessarily belong to the square type. It is somewhat similar to the identity matrix, however, in its case, not all components of the main diagonal take a value equal to unity. The main diagonal units can be two, four (it all depends on the length and width of the matrix). Or units may not exist at all (then it is considered zero). The remaining components of the canonical type, as well as the elements of the diagonal and unit ones, are equal to zero.
Triangular type
One of the most important types of matrix used in the search for its determinant and in performing simple operations. The triangular type comes from the diagonal, so the matrix is also square. The triangular form of the matrix is divided into upper triangular and lower triangular.
In the upper triangular matrix (Fig. 1), only the elements that are above the main diagonal take a value equal to zero. The components of the diagonal itself and the part of the matrix located below it contain numerical values.
In the lower triangular (Fig. 2), on the contrary, the elements located in the lower part of the matrix are equal to zero.
Stepped matrix
The view is necessary for finding the rank of the matrix, as well as for elementary actions on them (along with the triangular type). The step matrix is named so because it contains the characteristic "steps" of zeros (as shown in the figure). In the stepped type, a diagonal of zeros is formed (not necessarily the main one), and all elements under this diagonal also have values equal to zero. A prerequisite is the following: if a zero row is present in the step matrix, then the remaining rows below it also do not contain numerical values.
Thus, we examined the most important types of matrices needed to work with them. Now we will deal with the task of transforming the matrix into the required form.
Triangularization
How to bring the matrix to a triangular form? Most often in tasks, you need to transform the matrix into a triangular form in order to find its determinant, otherwise called the determinant. Performing this procedure, it is extremely important to “save” the main diagonal of the matrix, because the determinant of a triangular matrix is exactly the product of the components of its main diagonal. I also recall alternative methods for finding the determinant. The square type determinant is found using special formulas. For example, you can use the triangle method. For other matrices, the decomposition method is used for row, column, or their elements. You can also apply the method of minors and algebraic complements of the matrix.
We will analyze in detail the process of bringing the matrix to a triangular form using some examples as examples.
Exercise 1
It is necessary to find the determinant of the presented matrix using the method of reducing it to a triangular form.
The matrix given to us is a third-order square matrix. Therefore, to convert it to a triangular shape, we need to vanish two components of the first column and one component of the second.
To bring it to a triangular form, we start the transformation from the lower left corner of the matrix - from the number 6. To turn it to zero, we multiply the first row by three and subtract it from the last row.
Important! The top row does not change, but remains the same as in the original matrix. It is not necessary to record a line four times the original. But the values of the rows whose components need to be set to zero are constantly changing.
Next, we will deal with the following value - the element of the second row of the first column, number 8. Multiply the first row by four and subtract it from the second row. Get zero.
Only the last value remains - the element of the third row of the second column. This is the number (-1). To nullify it, subtract the second from the first line.
Let's check:
detA = 2 x (-1) x 11 = -22.
So, the answer to the task: -22.
Task 2
It is necessary to find the determinant of the matrix by reducing it to a triangular form.
The presented matrix belongs to the square type and is a fourth-order matrix. Therefore, it is necessary to vanish three components of the first column, two components of the second column and one component of the third.
We begin by casting it from the element located in the lower left corner, from the number 4. We need to turn this number to zero. It is most convenient to do this by multiplying the top line by four, and then subtract it from the fourth. We write the result of the first stage of the transformation.
So, the component of the fourth row is zero. We pass to the first element of the third row, to the number 3. We perform a similar operation. We multiply the first row by three, subtract it from the third row and write the result.
Next, see the number 2 in the second line. Repeat the operation: multiply the top line by two and subtract it from the second.
We managed to nullify all the components of the first column of this square matrix, with the exception of the number 1 - the element of the main diagonal that does not require a transformation. Now it is important to preserve the resulting zeros, so we will perform conversions with rows, not columns. We pass to the second column of the presented matrix.
Again, start from the bottom — from the second column of the last row. This is the number (-7). However, in this case it is more convenient to start with the number (-1) - the element of the second column of the third row. To nullify it, subtract the second from the third line. Then we multiply the second line by seven and subtract it from the fourth. We got zero instead of the element located in the fourth row of the second column. Now let's move on to the third column.
In this column, we need to vanish only one number - 4. It is not difficult to do this: we simply add the third to the last row and see the zero we need.
After all the transformations we performed, we reduced the proposed matrix to a triangular form. Now, to find its determinant, you only need to multiply the resulting elements of the main diagonal. We get : detA = 1 x (-1) x (-4) x 40 = 160. Therefore, the solution is the number 160.
So, now the question of reducing the matrix to a triangular form will not complicate you.
Cast to Step
In elementary operations on matrices, a stepped view is less "demanded" than a triangular one. Most often it is used to find the rank of a matrix (i.e., the number of its nonzero rows) or to determine linearly dependent and independent rows. However, the stepped form of the matrix is more universal, as it is suitable not only for the square type, but also for everyone else.
To bring the matrix to a stepwise form, you first need to find its determinant. To do this, the above methods are suitable. The purpose of finding the determinant is as follows: to find out whether it can be transformed into a stepwise form of the matrix. If the determinant is greater than or less than zero, then you can safely proceed to the task. If it is equal to zero, the matrix cannot be reduced to a stepwise form. In this case, you need to check if there are errors in the record or in the matrix transformations. If there are no such inaccuracies, the task cannot be solved.
Let us consider how to bring the matrix to a stepwise form using several tasks as examples.
Task 1. Find the rank of this matrix table.
Before us is a third-order square matrix (3x3). We know that in order to find a rank it is necessary to bring it to a stepwise form. Therefore, first we need to find the determinant of the matrix. We use the triangle method: detA = (1 x 5 x 0) + (2 x 1 x 2) + (6 x 3 x 4) - (1 x 1 x 4) - (2 x 3 x 0) - (6 x 5 x 2) = 12.
The determinant = 12. It is greater than zero, which means that the matrix can be reduced to a stepwise form. We proceed to its transformations.
Let's start it with the element of the left column of the third row - the number 2. Multiply the top row by two and subtract it from the third. Thanks to this operation, both the element we need and the number 4 — the element of the second column of the third row — have vanished.
Next, we zero out the element of the second row of the first column - the number 3. To do this, multiply the top row by three and subtract it from the second.
We see that as a result of the reduction a triangular matrix was formed. In our case, the conversion cannot be continued, since the remaining components cannot be turned to zero.
So, we conclude that the number of rows containing numerical values in this matrix (or its rank) is 3. The answer to the task: 3.
Task 2. Determine the number of linearly independent rows of this matrix.
We need to find strings that cannot be set to zero by any transformations. In fact, we need to find the number of nonzero rows, or the rank of the presented matrix. To do this, we simplify it.
We see a matrix that does not belong to a square type. It measures 3x4. We start the cast also with the element of the lower left corner - the number (-1).
Add the first row to the third. Next, subtract the second from it to turn the number 5 to zero.
Its further transformations are impossible. So, we conclude that the number of linearly independent lines in it and the answer to the task is 3.
Now the reduction of the matrix to a stepped form is not an impossible task for you.
Using the examples of these tasks, we examined the reduction of the matrix to a triangular form and a stepwise form. In order to nullify the desired values of the matrix tables, in some cases it is necessary to show imagination and correctly convert their columns or rows. I wish you success in mathematics and in working with matrices!