Mathematics for dummies. Matrices and basic operations on them

This topic will cover operations such as adding and subtracting matrices, multiplying a matrix by a number, multiplying a matrix by a matrix, and transposing a matrix. All symbols used on this page are taken from the previous topic.

Addition and subtraction of matrices.

The sum of $A+B$ of matrices $A_(m\times n)=(a_(ij))$ and $B_(m\times n)=(b_(ij))$ is called matrix $C_(m\times n) =(c_(ij))$, where $c_(ij)=a_(ij)+b_(ij)$ for all $i=\overline(1,m)$ and $j=\overline(1,n) $.

A similar definition is introduced for the difference of matrices:

The difference between the $A-B$ matrices $A_(m\times n)=(a_(ij))$ and $B_(m\times n)=(b_(ij))$ is the matrix $C_(m\times n)=( c_(ij))$, where $c_(ij)=a_(ij)-b_(ij)$ for all $i=\overline(1,m)$ and $j=\overline(1,n)$.

Explanation for the entry $i=\overline(1,m)$: show\hide

The notation "$i=\overline(1,m)$" means that the parameter $i$ varies from 1 to m. For example, the notation $i=\overline(1,5)$ indicates that the parameter $i$ takes the values ​​1, 2, 3, 4, 5.

It is worth noting that addition and subtraction operations are defined only for matrices of the same size. In general, addition and subtraction of matrices are operations that are clear intuitively, because they essentially mean just the summation or subtraction of the corresponding elements.

Example No. 1

Three matrices are given:

$$ A=\left(\begin(array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \end(array) \right)\;\; B=\left(\begin(array) (ccc) 10 & -25 & 98 \\ 3 & 0 & -14 \end(array) \right); \;\; F=\left(\begin(array) (cc) 1 & 0 \\ -5 & 4 \end(array) \right). $$

Is it possible to find the matrix $A+F$? Find matrices $C$ and $D$ if $C=A+B$ and $D=A-B$.

Matrix $A$ contains 2 rows and 3 columns (in other words, the size of matrix $A$ is $2\times 3$), and matrix $F$ contains 2 rows and 2 columns. The sizes of the matrices $A$ and $F$ do not coincide, so we cannot add them, i.e. the $A+F$ operation is not defined for these matrices.

The sizes of the matrices $A$ and $B$ are the same, i.e. The matrix data contains an equal number of rows and columns, so the addition operation is applicable to them.

$$ C=A+B=\left(\begin(array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \end(array) \right)+ \left(\begin(array ) (ccc) 10 & -25 & 98 \\ 3 & 0 & -14 \end(array) \right)=\\= \left(\begin(array) (ccc) -1+10 & -2+( -25) & 1+98 \\ 5+3 & 9+0 & -8+(-14) \end(array) \right)= \left(\begin(array) (ccc) 9 & -27 & 99 \\ 8 & 9 & -22 \end(array) \right) $$

Let's find the matrix $D=A-B$:

$$ D=A-B=\left(\begin(array) (ccc) -1 & -2 & 1 \\ 5 & 9 & -8 \end(array) \right)- \left(\begin(array) ( ccc) 10 & -25 & 98 \\ 3 & 0 & -14 \end(array) \right)=\\= \left(\begin(array) (ccc) -1-10 & -2-(-25 ) & 1-98 \\ 5-3 & 9-0 & -8-(-14) \end(array) \right)= \left(\begin(array) (ccc) -11 & 23 & -97 \ \2 & 9 & 6 \end(array) \right) $$

Answer: $C=\left(\begin(array) (ccc) 9 & -27 & 99 \\ 8 & 9 & -22 \end(array) \right)$, $D=\left(\begin(array) (ccc) -11 & 23 & -97 \\ 2 & 9 & 6 \end(array) \right)$.

Multiplying a matrix by a number.

The product of the matrix $A_(m\times n)=(a_(ij))$ by the number $\alpha$ is the matrix $B_(m\times n)=(b_(ij))$, where $b_(ij)= \alpha\cdot a_(ij)$ for all $i=\overline(1,m)$ and $j=\overline(1,n)$.

Simply put, multiplying a matrix by a certain number means multiplying each element of a given matrix by that number.

Example No. 2

The matrix is ​​given: $ A=\left(\begin(array) (ccc) -1 & -2 & 7 \\ 4 & 9 & 0 \end(array) \right)$. Find matrices $3\cdot A$, $-5\cdot A$ and $-A$.

$$ 3\cdot A=3\cdot \left(\begin(array) (ccc) -1 & -2 & 7 \\ 4 & 9 & 0 \end(array) \right) =\left(\begin( array) (ccc) 3\cdot(-1) & 3\cdot(-2) & 3\cdot 7 \\ 3\cdot 4 & 3\cdot 9 & 3\cdot 0 \end(array) \right)= \left(\begin(array) (ccc) -3 & -6 & 21 \\ 12& 27 & 0 \end(array) \right).\\ -5\cdot A=-5\cdot \left(\begin (array) (ccc) -1 & -2 & 7 \\ 4 & 9 & 0 \end(array) \right) =\left(\begin(array) (ccc) -5\cdot(-1) & - 5\cdot(-2) & -5\cdot 7 \\ -5\cdot 4 & -5\cdot 9 & -5\cdot 0 \end(array) \right)= \left(\begin(array) ( ccc) 5 & 10 & -35 \\ -20 & -45 & 0 \end(array) \right). $$

The notation $-A$ is a shorthand notation for $-1\cdot A$. That is, to find $-A$ you need to multiply all elements of the matrix $A$ by (-1). Essentially, this means that the sign of all elements of the matrix $A$ will change to the opposite:

$$ -A=-1\cdot A=-1\cdot \left(\begin(array) (ccc) -1 & -2 & 7 \\ 4 & 9 & 0 \end(array) \right)= \ left(\begin(array) (ccc) 1 & 2 & -7 \\ -4 & -9 & 0 \end(array) \right) $$

Answer: $3\cdot A=\left(\begin(array) (ccc) -3 & -6 & 21 \\ 12& 27 & 0 \end(array) \right);\; -5\cdot A=\left(\begin(array) (ccc) 5 & 10 & -35 \\ -20 & -45 & 0 \end(array) \right);\; -A=\left(\begin(array) (ccc) 1 & 2 & -7 \\ -4 & -9 & 0 \end(array) \right)$.

Product of two matrices.

The definition of this operation is cumbersome and, at first glance, unclear. Therefore, first I will indicate a general definition, and then we will analyze in detail what it means and how to work with it.

The product of the matrix $A_(m\times n)=(a_(ij))$ by the matrix $B_(n\times k)=(b_(ij))$ is the matrix $C_(m\times k)=(c_( ij))$, for which each element $c_(ij)$ is equal to the sum of the products of the corresponding elements of the i-th rows of matrix $A$ to elements of the j-th column of matrix $B$: $$c_(ij)=\sum\limits_(p=1)^(n)a_(ip)b_(pj), \;\; i=\overline(1,m), j=\overline(1,n).$$

Let's look at matrix multiplication step by step using an example. However, you should immediately note that not all matrices can be multiplied. If we want to multiply matrix $A$ by matrix $B$, then we first need to make sure that the number of columns of matrix $A$ is equal to the number of rows of matrix $B$ (such matrices are often called agreed upon). For example, the matrix $A_(5\times 4)$ (the matrix contains 5 rows and 4 columns) cannot be multiplied by the matrix $F_(9\times 8)$ (9 rows and 8 columns), since the number of columns of the matrix $A $ is not equal to the number of rows of the matrix $F$, i.e. $4\neq 9$. But you can multiply the matrix $A_(5\times 4)$ by the matrix $B_(4\times 9)$, since the number of columns of the matrix $A$ is equal to the number of rows of the matrix $B$. In this case, the result of multiplying the matrices $A_(5\times 4)$ and $B_(4\times 9)$ will be the matrix $C_(5\times 9)$, containing 5 rows and 9 columns:

Example No. 3

Given matrices: $ A=\left(\begin(array) (cccc) -1 & 2 & -3 & 0 \\ 5 & 4 & -2 & 1 \\ -8 & 11 & -10 & -5 \end (array) \right)$ and $ B=\left(\begin(array) (cc) -9 & 3 \\ 6 & 20 \\ 7 & 0 \\ 12 & -4 \end(array) \right) $. Find the matrix $C=A\cdot B$.

First, let's immediately determine the size of the matrix $C$. Since matrix $A$ has size $3\times 4$, and matrix $B$ has size $4\times 2$, then the size of matrix $C$ is: $3\times 2$:

So, as a result of the product of matrices $A$ and $B$, we should obtain a matrix $C$, consisting of three rows and two columns: $ C=\left(\begin(array) (cc) c_(11) & c_( 12) \\ c_(21) & c_(22) \\ c_(31) & c_(32) \end(array) \right)$. If the designations of elements raise questions, then you can look at the previous topic: “Matrixes. Types of matrices. Basic terms”, at the beginning of which the designation of matrix elements is explained. Our goal: find the values ​​of all elements of the matrix $C$.

Let's start with the element $c_(11)$. To obtain the element $c_(11)$, you need to find the sum of the products of the elements of the first row of the matrix $A$ and the first column of the matrix $B$:

To find the element $c_(11)$ itself, you need to multiply the elements of the first row of the matrix $A$ by the corresponding elements of the first column of the matrix $B$, i.e. the first element to the first, the second to the second, the third to the third, the fourth to the fourth. We summarize the results obtained:

$$ c_(11)=-1\cdot (-9)+2\cdot 6+(-3)\cdot 7 + 0\cdot 12=0. $$

Let's continue the solution and find $c_(12)$. To do this, you will have to multiply the elements of the first row of matrix $A$ and the second column of matrix $B$:

Similar to the previous one, we have:

$$ c_(12)=-1\cdot 3+2\cdot 20+(-3)\cdot 0 + 0\cdot (-4)=37. $$

All elements of the first row of matrix $C$ have been found. Let's move on to the second line, which begins with the element $c_(21)$. To find it, you will have to multiply the elements of the second row of matrix $A$ and the first column of matrix $B$:

$$ c_(21)=5\cdot (-9)+4\cdot 6+(-2)\cdot 7 + 1\cdot 12=-23. $$

We find the next element $c_(22)$ by multiplying the elements of the second row of matrix $A$ by the corresponding elements of the second column of matrix $B$:

$$ c_(22)=5\cdot 3+4\cdot 20+(-2)\cdot 0 + 1\cdot (-4)=91. $$

To find $c_(31)$, multiply the elements of the third row of matrix $A$ by the elements of the first column of matrix $B$:

$$ c_(31)=-8\cdot (-9)+11\cdot 6+(-10)\cdot 7 + (-5)\cdot 12=8. $$

And finally, to find the element $c_(32)$, you will have to multiply the elements of the third row of the matrix $A$ by the corresponding elements of the second column of the matrix $B$:

$$ c_(32)=-8\cdot 3+11\cdot 20+(-10)\cdot 0 + (-5)\cdot (-4)=216. $$

All elements of the matrix $C$ have been found, all that remains is to write that $C=\left(\begin(array) (cc) 0 & 37 \\ -23 & 91 \\ 8 & 216 \end(array) \right)$ . Or, to write in full:

$$ C=A\cdot B =\left(\begin(array) (cccc) -1 & 2 & -3 & 0 \\ 5 & 4 & -2 & 1 \\ -8 & 11 & -10 & - 5 \end(array) \right)\cdot \left(\begin(array) (cc) -9 & 3 \\ 6 & 20 \\ 7 & 0 \\ 12 & -4 \end(array) \right) =\left(\begin(array) (cc) 0 & 37 \\ -23 & 91 \\ 8 & 216 \end(array) \right). $$

Answer: $C=\left(\begin(array) (cc) 0 & 37 \\ -23 & 91 \\ 8 & 216 \end(array) \right)$.

By the way, there is often no reason to describe in detail the location of each element of the result matrix. For matrices whose size is small, you can do this:

$$ \left(\begin(array) (cc) 6 & 3 \\ -17 & -2 \end(array)\right)\cdot \left(\begin(array) (cc) 4 & 9 \\ - 6 & 90 \end(array) \right) =\left(\begin(array) (cc) 6\cdot(4)+3\cdot(-6) & 6\cdot(9)+3\cdot(90 ) \\ -17\cdot(4)+(-2)\cdot(-6) & -17\cdot(9)+(-2)\cdot(90) \end(array) \right) =\left (\begin(array) (cc) 6 & 324 \\ -56 & -333 \end(array) \right) $$

It is also worth noting that matrix multiplication is non-commutative. This means that in general case$A\cdot B\neq B\cdot A$. Only for some types of matrices, which are called permutable(or commuting), the equality $A\cdot B=B\cdot A$ is true. It is precisely based on the non-commutativity of multiplication that we need to indicate exactly how we multiply the expression by a particular matrix: on the right or on the left. For example, the phrase “multiply both sides of the equality $3E-F=Y$ by the matrix $A$ on the right” means that you want to get the following equality: $(3E-F)\cdot A=Y\cdot A$.

Transposed with respect to the matrix $A_(m\times n)=(a_(ij))$ is the matrix $A_(n\times m)^(T)=(a_(ij)^(T))$, for elements which $a_(ij)^(T)=a_(ji)$.

Simply put, in order to obtain a transposed matrix $A^T$, you need to replace the columns in the original matrix $A$ with the corresponding rows according to this principle: there was a first row - there will be a first column; there was a second row - there will be a second column; there was a third row - there will be a third column and so on. For example, let's find the transposed matrix to the matrix $A_(3\times 5)$:

Accordingly, if the original matrix had a size of $3\times 5$, then the transposed matrix has a size of $5\times 3$.

Some properties of operations on matrices.

Here it is assumed that $\alpha$, $\beta$ are some numbers, and $A$, $B$, $C$ are matrices. For the first four properties I indicated names; the rest can be named by analogy with the first four.

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, what if in simple language– table of numbers.

Typically matrices are denoted in capitals with 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 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 jth column, will be equal to the sum of the products of the corresponding elements in i-th line 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 cope.

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.

Multiplying a matrix by a number is an operation on a matrix, as a result of which each of its elements is multiplied by a real or complex number. It looks like this in mathematical language:

$$ B = \lambda \cdot A \Rightarrow b_(ij) = \lambda a_(ij) $$

It is worth noting that the resulting matrix $ B $ should have the same dimension as the initial matrix $ A $. You can also pay attention to the following fact: $ \lambda \cdot A = A \cdot \lambda $, that is, you can swap the factors and this will not change the product.

It will be useful to use the operation of multiplying a matrix by a number when moving the common factor outside the matrix. In this case, each element of the matrix is ​​divided by the number $ \lambda $, and it itself is placed in front of the matrix.

Properties

1. Distributive law regarding matrices: $$ \lambda \cdot (A+B) = \lambda A + \lambda B $$Multiplying the sum of matrices by a number can be replaced by the sum of the products of each individual matrix by a given number
2. Distributive law for real (complex) numbers: $$ (\lambda + \mu) \cdot A = \lambda A + \mu A $$ Multiplying a matrix by the sum of numbers can be replaced by the sum of the products of each number by the matrix
3. Associative law: $$ \lambda \cdot (\mu \cdot A) = (\lambda \cdot \mu) A $$ Convenient to use if you need to remove the common factor from the matrix in front of it, while multiplying the coefficient already in front of it
4. There is a special number $ \lambda = 1 $, thanks to which the matrix remains unchanged $$ 1 \cdot A = A \cdot 1 = A $$
5. Multiplying a matrix by zero leads to the fact that each element of the matrices is zeroed and the matrix becomes zero of the same dimension as it was initially: $$ 0 \cdot A = 0 $$

Examples of solutions

In order to multiply matrix A by an arbitrary number α, you need matrix elements A multiply by the number α, i.e. the product of the matrix and the number will be as follows:

Example 1. Find matrix 3 A for matrix

Solution. In accordance with the definition, we multiply the elements of the matrix A by 3 and we get

This was a very simple example of multiplying a matrix by a number with integers. Ahead also simple examples, but already such, where among the factors and elements of matrices there are fractions, variables ( letter designations), because the laws of multiplication apply not only to integers, so it never hurts to repeat them.

Example 2. A by the number α, if
, .

A by α, not forgetting that when multiplying fractions, the numerator of the first fraction is multiplied by the numerator of the first fraction and the product is written in the numerator, and the denominator of the first fraction is multiplied by the denominator of the second fraction and the product is written in the denominator. When receiving the second element of the first row of the new matrix, the resulting fraction was reduced by 2; this must be done. We get

Example 3. Perform matrix multiplication operation A by the number α, if
, .

Solution. Let's multiply the matrix elements A by α, without getting confused in the letter designations, not forgetting to leave a minus before the second element of the second row of the new matrix, and remembering that the result of multiplying a number by its inverse is one (the first element of the third row). We get

.

Example 4. Perform matrix multiplication operation A by the number α, if
, .

Solution. Remember that when you multiply a number to a power by a number to a power, the exponents add up. We get

.

This example, among other things, clearly demonstrates that the actions of multiplying a matrix by a number can be read (and written) in reverse order and this is called placing a constant factor in front of the matrix.

In combination with adding and subtracting matrices the operation of multiplying a matrix by a number can produce various matrix expressions, for example, 5 A − 3B , 4A + 2B .

Properties of multiplying a matrix by a number

(here A, B are matrices, are numbers, 1 is the number one)

1.

2.

3.

Properties (1) and (2) connect multiplication of a matrix by a number with addition of matrices. There is also a very important connection between multiplying a matrix by a number and multiplying matrices themselves:

that is, if in a product of matrices one of the factors is multiplied by a number, then the entire product will be multiplied by the number.

Lecture No. 1

MATRICES

Definition and types of matrices

Definition 1.1.Matrix size T P is a rectangular table of numbers (or other objects) containing m lines and n columns.

Matrices are denoted by capital letters of the Latin alphabet, for example, A, B, C,... The numbers (or other objects) that make up a matrix are called elements matrices. Matrix elements can be functions. To designate matrix elements, lowercase letters of the Latin alphabet with double indexing are used: аij, where is the first index i(read – and) – line number, second index j(read – zhi) – column number.

Definition 1.2. The matrix is ​​called square n- first order if the number of its rows is equal to the number of columns and equal to the same number P

For a square matrix, the concepts are introduced main and secondary diagonals.

Definition 1.3.Main diagonal a square matrix consists of elements having the same indices, i.e. . These are the elements: a 11,a 22,…

Definition 1.4. diagonal, if all elements except those on the main diagonal are zero

Definition 1.5. The square matrix is ​​called triangular, if all its elements located below (or above) the main diagonal are equal to zero.

Definition 1.6. Square matrix P- of order, in which all elements of the main diagonal are equal to one, and the rest are equal to zero, is called single matrix n-th order, and it is denoted by the letter E.

Definition 1.7. A matrix of any size is called null, or null matrix, if all its elements are equal to zero.

Definition 1.8. A matrix consisting of one row is called row matrix.

Definition 1.9. A matrix consisting of one column is called matrix-column.

A = (a 11 A 12 ... A 1n) – matrix-row;

Definition 1.10. Two matrices A And IN identical sizes are called equal if all corresponding elements of these matrices are equal to each other, i.e. aij = bij for any i= 1, 2, ..., T; j = 1, 2,…, n.

Operations on matrices

A number of operations can be performed on matrices, as well as on numbers. The main operations on matrices are addition (subtraction) of matrices, multiplication of a matrix by a number, multiplication of matrices. These operations are similar to operations on numbers. A specific operation is matrix transposition.

Multiplying a matrix by a number

Definition 1.11.Product of matrix A by numberλ is called a matrix B = A, whose elements are obtained by multiplying the elements of the matrix A by the number λ .

Example 1.1. Find matrix product A= to the number 5.

Solution. .◄ 5A=

Rule for multiplying a matrix by a number: To multiply a matrix by a number, you need to multiply all the elements of the matrix by that number.

Consequence.

1. The common factor of all matrix elements can be taken out of the matrix sign.

2. Matrix product A for the number 0 there is a zero matrix: A· 0 = 0 .

Matrix addition

Definition 1.12.The sum of two matrices A and B same size t n called a matrix WITH= A+ IN, the elements of which are obtained by adding the corresponding elements of the matrix A and matrices IN, i.e. cij = aij + bij For i = 1, 2, ..., m; j= 1, 2, ..., n(i.e., the matrices are added element by element).

Consequence. Sum matrix A with a zero matrix is ​​equal to the original matrix: A + O = A.

1.2.3. Subtraction of matrices

Difference of two matrices of the same size is determined through the previous operations: A – B = A + (– 1)IN.

Definition 1.13. Matrix –A = (– 1)A called opposite matrix A.

Consequence. The sum of opposite matrices is equal to the zero matrix : A + (–A) = O.

Matrix multiplication

Definition 1.14.Multiplying matrix A by matrix B defined when the number of columns of the first matrix is ​​equal to the number of rows of the second matrix. Then product of matrices such a matrix is ​​called , each element of which cij equal to the sum of the products of the elements i th row of the matrix A to the corresponding elements j th matrix column B.

Example 1.4. Calculate matrix product A · B, Where

A=

=

Example 1.5. Find matrix products AB And VA, Where

Notes. From examples 1.4–1.5 it follows that the operation of matrix multiplication has some differences from multiplication of numbers:

1) if the product of matrices AB exists, then after rearranging the factors the product of the matrices VA may not exist. Indeed, in Example 1.4 the matrix product AB exists, but the matrix product BA does not exist;

2) even if the works AB And VA exist, then the result of the product can be matrices of different sizes. In the case when both works AB And VA there exist both matrices of the same size (this is only possible when multiplying square matrices of the same order), then the commutative (commutative) law of multiplication still does not hold, those. A B In A, as in example 1.5;

3) however, if you multiply the square matrix A to the identity matrix E of the same order, then AE = EA = A.

Thus, the identity matrix plays the same role in matrix multiplication as the number 1 does in number multiplication;

4) the product of two non-zero matrices can be equal to the zero matrix, i.e. from the fact that A B= 0, it does not follow that A = 0 or B= 0.