Matrix addition:
Subtraction and addition of matrices reduces to the corresponding operations on their elements. Matrix addition operation entered only for matrices the same size, i.e. for matrices, in which the number of rows and columns is respectively equal. Sum of matrices A and B are called matrix C, whose elements are equal to the sum of the corresponding elements. C = A + B c ij = a ij + b ij Defined similarly matrix difference.
Multiplying a matrix by a number:
Matrix multiplication (division) operation of any size by an arbitrary number is reduced to multiplying (dividing) each element matrices for this number. Matrix product And the number k is called matrix B, such that
b ij = k × a ij . B = k × A b ij = k × a ij . Matrix- A = (-1) × A is called the opposite matrix A.
Properties of adding matrices and multiplying a matrix by a number:
Matrix addition operations And matrix multiplication on a number have the following properties: 1. A + B = B + A; 2. A + (B + C) = (A + B) + C; 3. A + 0 = A; 4. A - A = 0; 5. 1 × A = A; 6. α × (A + B) = αA + αB; 7. (α + β) × A = αA + βA; 8. α × (βA) = (αβ) × A; , where A, B and C are matrices, α and β are numbers.
Matrix multiplication (Matrix product):
Operation of multiplying two matrices is entered only for the case when the number of columns of the first matrices equal to the number of lines of the second matrices. Matrix product And m×n on matrix In n×p, called matrix With m×p such that with ik = a i1 × b 1k + a i2 × b 2k + ... + a in × b nk , i.e., the sum of the products of the elements of the i-th row is found matrices And to the corresponding elements of the jth column matrices B. If matrices A and B are squares of the same size, then the products AB and BA always exist. It is easy to show that A × E = E × A = A, where A is square matrix, E - unit matrix the same size.
Properties of matrix multiplication:
Matrix multiplication not commutative, i.e. AB ≠ BA even if both products are defined. However, if for any matrices the relationship AB=BA is satisfied, then such matrices are called commutative. The most typical example is a single matrix, which commutes with any other matrix the same size. Only square ones can be permutable matrices of the same order. A × E = E × A = A
Matrix multiplication has the following properties: 1. A × (B × C) = (A × B) × C; 2. A × (B + C) = AB + AC; 3. (A + B) × C = AC + BC; 4. α × (AB) = (αA) × B; 5. A × 0 = 0; 0 × A = 0; 6. (AB) T = B T A T; 7. (ABC) T = C T V T A T; 8. (A + B) T = A T + B T;
2. Determinants of the 2nd and 3rd orders. Properties of determinants.
Matrix determinant second order, or determinant second order is a number that is calculated by the formula:
Matrix determinant third order, or determinant third order is a number that is calculated by the formula:
This number represents an algebraic sum consisting of six terms. Each term contains exactly one element from each row and each column matrices. Each term consists of the product of three factors.
Signs with which members determinant of the matrix included in the formula finding the determinant of the matrix third order can be determined using the given scheme, which is called the rule of triangles or Sarrus's rule. The first three terms are taken with a plus sign and determined from the left figure, and the next three terms are taken with a minus sign and determined from the right figure.
Determine the number of terms to find determinant of the matrix, in an algebraic sum, you can calculate the factorial: 2! = 1 × 2 = 2 3! = 1 × 2 × 3 = 6
Properties of matrix determinants
Properties of matrix determinants:
Property #1:
Matrix determinant will not change if its rows are replaced with columns, each row with a column with the same number, and vice versa (Transposition). |A| = |A| T
Consequence:
Columns and Rows determinant of the matrix are equal, therefore, the properties inherent in rows also apply to columns.
Property #2:
When rearranging 2 rows or columns matrix determinant will change the sign to the opposite one, maintaining the absolute value, i.e.:
Property #3:
Matrix determinant having two identical rows is equal to zero.
Property #4:
Common factor of elements of any series determinant of the matrix can be taken as a sign determinant.
Corollaries from properties No. 3 and No. 4:
If all elements of a certain series (row or column) are proportional to the corresponding elements of a parallel series, then such matrix determinant equal to zero.
Property #5:
determinant of the matrix are equal to zero, then matrix determinant equal to zero.
Property #6:
If all elements of a row or column determinant presented as a sum of 2 terms, then determinant matrices can be represented as the sum of 2 determinants according to the formula:
Property #7:
If to any row (or column) determinant add the corresponding elements of another row (or column), multiplied by the same number, then matrix determinant will not change its value.
Example of using properties for calculation determinant of the matrix:
This manual will help you learn how to perform operations with matrices: addition (subtraction) of matrices, transposition of a matrix, multiplication of matrices, finding the inverse matrix. All material is presented in a simple and accessible form, relevant examples are given, so even an unprepared person can learn how to perform actions with matrices. For self-monitoring and self-testing, you can download a matrix calculator for free >>>.
I will try to minimize theoretical calculations, in some places explanations “on the fingers” and the use of non-scientific terms are possible. Lovers of solid theory, please do not engage in criticism, our task is learn to perform operations with matrices.
For SUPER FAST preparation on the topic (who is “on fire”) there is an intensive pdf course Matrix, determinant and test!
A matrix is a rectangular table of some elements. As elements we will consider numbers, that is, numerical matrices. ELEMENT is a term. It is advisable to remember the term, it will appear often, it is no coincidence that I used bold font to highlight it.
Designation: matrices are usually denoted in capital Latin letters
Example: Consider a two-by-three matrix:
This matrix consists of six elements:
All numbers (elements) inside the matrix exist on their own, that is, there is no question of any subtraction: 
It's just a table (set) of numbers!
We'll also agree don't rearrange numbers, unless otherwise stated in the explanations. Each number has its own location and cannot be shuffled!
The matrix in question has two rows: 
and three columns: 
STANDARD: when talking about matrix sizes, then at first indicate the number of rows, and only then the number of columns. We have just broken down the two-by-three matrix.
If the number of rows and columns of a matrix is the same, then the matrix is called square, For example: 
– a three-by-three matrix.
If a matrix has one column or one row, then such matrices are also called vectors.
In fact, we have known the concept of a matrix since school; consider, for example, a point with coordinates “x” and “y”: . Essentially, the coordinates of a point are written into a one-by-two matrix. By the way, here is an example of why the order of numbers matters: and are two completely different points on the plane.
Now let's move on to studying operations with matrices:
1) Act one. Removing a minus from the matrix (introducing a minus into the matrix).
Let's return to our matrix 
. As you probably noticed, there are too many negative numbers in this matrix. This is very inconvenient from the point of view of performing various actions with the matrix, it is inconvenient to write so many minuses, and it simply looks ugly in design.
Let's move the minus outside the matrix by changing the sign of EACH element of the matrix:
At zero, as you understand, the sign does not change; zero is also zero in Africa.
Reverse example: 
. It looks ugly.
Let's introduce a minus into the matrix by changing the sign of EACH element of the matrix:
Well, it turned out much nicer. And, most importantly, it will be EASIER to perform any actions with the matrix. Because there is such a mathematical folk sign: the more minuses, the more confusion and errors.
2) Act two. Multiplying a matrix by a number.
Example:
It's simple, in order to multiply a matrix by a number, you need every matrix element multiplied by a given number. In this case - a three.
Another useful example:
– multiplying a matrix by a fraction
First let's look at what to do NO NEED:
There is NO NEED to enter a fraction into the matrix; firstly, it only complicates further actions with the matrix, and secondly, it makes it difficult for the teacher to check the solution (especially if 
– final answer of the task).
And, moreover, NO NEED divide each element of the matrix by minus seven:
From the article Mathematics for dummies or where to start, we remember that in higher mathematics they try to avoid decimal fractions with commas in every possible way.
The only thing is preferably What to do in this example is to add a minus to the matrix:
But if only ALL matrix elements were divided by 7 without a trace, then it would be possible (and necessary!) to divide.
Example:
In this case, you can NEED TO multiply all matrix elements by , since all matrix numbers are divisible by 2 without a trace.
Note: in the theory of higher school mathematics there is no concept of “division”. Instead of saying “this divided by that,” you can always say “this multiplied by a fraction.” That is, division is a special case of multiplication.
3) Act three. Matrix Transpose.
In order to transpose a matrix, you need to write its rows into the columns of the transposed matrix.
Example:
Transpose matrix
There is only one line here and, according to the rule, it needs to be written in a column:
– transposed matrix.
A transposed matrix is usually indicated by a superscript or a prime at the top right.
Step by step example:
Transpose matrix 
First we rewrite the first row into the first column:
Then we rewrite the second line into the second column: 
And finally, we rewrite the third row into the third column:
Ready. Roughly speaking, transposing means turning the matrix on its side.
4) Act four. Sum (difference) of matrices.
The sum of matrices is a simple operation.
NOT ALL MATRICES CAN BE FOLDED. To perform addition (subtraction) of matrices, it is necessary that they be the SAME SIZE.
For example, if a two-by-two matrix is given, then it can only be added with a two-by-two matrix and no other! 
Example:
Add matrices 
And 
In order to add matrices, you need to add their corresponding elements:
For the difference of matrices the rule is similar, it is necessary to find the difference of the corresponding elements.
Example:
Find matrix difference 
, 
How can you solve this example more easily, so as not to get confused? It is advisable to get rid of unnecessary minuses; to do this, add a minus to the matrix:
Note: in the theory of higher school mathematics there is no concept of “subtraction”. Instead of saying “subtract this from this,” you can always say “add a negative number to this.” That is, subtraction is a special case of addition.
5) Act five. Matrix multiplication.
What matrices can be multiplied?
In order for a matrix to be multiplied by a matrix, it is necessary so that the number of matrix columns is equal to the number of matrix rows.
Example:
Is it possible to multiply a matrix by a matrix?
This means that matrix data can be multiplied.
But if the matrices are rearranged, then, in this case, multiplication is no longer possible!
Therefore, multiplication is not possible:
It is not so rare to encounter tasks with a trick, when the student is asked to multiply matrices, the multiplication of which is obviously impossible.
It should be noted that in some cases it is possible to multiply matrices in both ways.
For example, for matrices, and both multiplication and multiplication are possible
1st year, higher mathematics, studying matrices and basic actions on them. Here we systematize the basic operations that can be performed with matrices. Where to start getting acquainted with matrices? Of course, from the simplest things - definitions, basic concepts and simple operations. We assure you that the matrices will be understood by everyone who devotes at least a little time to them!
Matrix Definition
Matrix is a rectangular table of elements. Well, in simple terms – a table of numbers.
Typically, matrices are denoted in capital Latin letters. For example, matrix A , matrix B and so on. Matrices can be of different sizes: rectangular, square, and there are also row and column matrices called vectors. The size of the matrix is determined by the number of rows and columns. For example, let's write a rectangular matrix of size m on n , Where m – number of lines, and n – number of columns.
Items for which i=j (a11, a22, .. ) form the main diagonal of the matrix and are called diagonal.
What can you do with matrices? Add/Subtract, multiply by a number, multiply among themselves, transpose. Now about all these basic operations on matrices in order.
Matrix addition and subtraction operations
Let us immediately warn you that you can only add matrices of the same size. The result will be a matrix of the same size. Adding (or subtracting) matrices is simple - you just need to add up their corresponding elements . Let's give an example. Let's perform the addition of two matrices A and B of size two by two.
Subtraction is performed by analogy, only with the opposite sign.
Any matrix can be multiplied by an arbitrary number. To do this you need to multiply each of its elements by this number. For example, let's multiply the matrix A from the first example by the number 5:
Matrix multiplication operation
Not all matrices can be multiplied together. For example, we have two matrices - A and B. They can be multiplied by each other only if the number of columns of matrix A is equal to the number of rows of matrix B. In this case each element of the resulting matrix, located in the i-th row and j-th column, will be equal to the sum of the products of the corresponding elements in the i-th row of the first factor and the j-th column of the second. To understand this algorithm, let's write down how two square matrices are multiplied:
And an example with real numbers. Let's multiply the matrices:
Matrix transpose operation
Matrix transposition is an operation where the corresponding rows and columns are swapped. For example, let's transpose the matrix A from the first example:
Matrix determinant
Determinant, or determinant, is one of the basic concepts of linear algebra. Once upon a time, people came up with linear equations, and after them they had to come up with a determinant. In the end, it’s up to you to deal with all this, so, the last push!
The determinant is a numerical characteristic of a square matrix, which is needed to solve many problems.
To calculate the determinant of the simplest square matrix, you need to calculate the difference between the products of the elements of the main and secondary diagonals.
The determinant of a matrix of first order, that is, consisting of one element, is equal to this element.
What if the matrix is three by three? This is more difficult, but you can manage it.
For such a matrix, the value of the determinant is equal to the sum of the products of the elements of the main diagonal and the products of the elements lying on the triangles with a face parallel to the main diagonal, from which the product of the elements of the secondary diagonal and the product of the elements lying on the triangles with the face of the parallel secondary diagonal are subtracted.
Fortunately, in practice it is rarely necessary to calculate determinants of matrices of large sizes.
Here we looked at basic operations on matrices. Of course, in real life you may never encounter even a hint of a matrix system of equations, or, on the contrary, you may encounter much more complex cases when you really have to rack your brains. It is for such cases that professional student services exist. Ask for help, get a high-quality and detailed solution, enjoy academic success and free time.
