Instruction

1

To find change of any quantity, measure or calculate its initial value (x1).

2

Calculate or measure the final value of the same magnitude (x2).

3

Way to change this value according to the formula: Δx=x2-x1. For example: the initial value of the electric circuit voltage U1=220 V and the final value U2=120V. The change in voltage (or Delta voltage) will be equal to ΔU=U2–U1=220V-120V=100V

4

For finding the absolute error of measurement determine the exact or as it is sometimes called, the true value of any quantity (x0).

5

Take the approximate (measured – measured) value of the same variable (x).

6

Find the absolute error of measurement according to the formula: Δx=|x-x0|. For example: the exact number of residents - 8253 resident (x0=8253), rounding that number to 8300 (approximate value of x=8300). The absolute error (or Delta x) will be equal to Δx=|8300-8253|=47, and rounding up to 8200 (x=8200), the absolute error Δx=|8200-8253|=53. Thus, rounding to the number 8300 will be more accurate.

7

To compare the values of the function F(x) in a strictly fixed point x0 with the values of this function at any other point of x lying in the vicinity of x0, it uses the concept of "increment function" (ΔF) and the "increment of the argument of the function" (Δx). Δx is sometimes called the "increment of the independent variable". Find the increment of the argument by the formula Δx=x-x0.

8

Determine the function values at the points x0 and x and denote them respectively F(x0) and F(x).

9

Calculate increment of the function: ΔF= F(x)- F(x0). Example: to find the increment of the argument and increment of a function F(x)=x2+1 when you change the argument from 2 to 3. In this case, x0 is equal to 2, and x=3.

The increment of the argument (or Delta x) is Δx=3-2=1.

F(x0)= х02+1= 22+1=5.

F(x)= x2+1= 32+1=10.

The increment of the function (or Delta EF) ΔF= F(x)- F(x0)=10-5=5

The increment of the argument (or Delta x) is Δx=3-2=1.

F(x0)= х02+1= 22+1=5.

F(x)= x2+1= 32+1=10.

The increment of the function (or Delta EF) ΔF= F(x)- F(x0)=10-5=5

Note

You need not subtract a larger number from a smaller, and from the final value (not important: it more or less) elementary!

Useful advice

When finding all values of Δ do not use the same unit of measure.

# Advice 2 : How to count matrix

The term "matrix" is known from linear algebra. Before describing the permissible operations on matrices, it is necessary to introduce its definition. A matrix is a rectangular table of numbers containing some number m of rows and some n columns. If m = n, then the matrix is called square.

**Matrix**is usually denoted in capital Latin letters, e.g. A, or A = (aij), where (aij) is the matrix element of the i – row number j – column number. Let the two matrices A = (aij) and B = (bij) have the same dimension m*n.Instruction

1

The sum of matrices A = (aij) and B = (bij) is the matrix C = (cij) of the same dimension , where the elements cij determined by the equation cij = aij + bij (i = 1, 2, ..., m; j = 1, 2,..., n).

Addition of matrices has the following properties:

1. A + B = B + A

2. (A + B) + C = A + (B + C)

Addition of matrices has the following properties:

1. A + B = B + A

2. (A + B) + C = A + (B + C)

2

The product of the matrix A = (aij) a real number ? is the matrix C = (cij), where the elements cij determined by the equation cij = ? * aij (i = 1, 2, ..., m; j = 1, 2,..., n).

Multiplication of matrix by a number has the following properties:

1. (??)A = ?(?A)? and ? – real numbers,

2. ?(A + B) = ?A + ?In ? is a real number,

3. (? + ?)= ?+ ?In ? and ? – real numbers.

Introducing multiplication of a matrix by a scalar, we can introduce the operation of subtraction of matrices. The difference between the matrices A and B is matrix C, which can be calculated according to the rule:

C = A + (-1)*B

Multiplication of matrix by a number has the following properties:

1. (??)A = ?(?A)? and ? – real numbers,

2. ?(A + B) = ?A + ?In ? is a real number,

3. (? + ?)= ?+ ?In ? and ? – real numbers.

Introducing multiplication of a matrix by a scalar, we can introduce the operation of subtraction of matrices. The difference between the matrices A and B is matrix C, which can be calculated according to the rule:

C = A + (-1)*B

3

The product of matrices. The matrix A can be multiplied by a matrix B if the number of columns of matrix A equals the number of rows of the matrix B.

The product of the matrix A = (aij) of dimension m*n matrix B = (bij) of dimension n*p is the matrix C = (cij) is of dimension m*p, where the elements cij is given by cij = ai1*b1j + ai2*b2j + ... + ain*bnj (i = 1, 2, ..., m; j = 1, 2 ..., p).

The figure shows an example of matrix multiplication of dimension 2*2.

The product of matrices has the following properties:

1. (A * B) * C = A * (B * C)

2. (A + B) * C = A*C + B*C or A * (B + C) = A*B + A*C

The product of the matrix A = (aij) of dimension m*n matrix B = (bij) of dimension n*p is the matrix C = (cij) is of dimension m*p, where the elements cij is given by cij = ai1*b1j + ai2*b2j + ... + ain*bnj (i = 1, 2, ..., m; j = 1, 2 ..., p).

The figure shows an example of matrix multiplication of dimension 2*2.

The product of matrices has the following properties:

1. (A * B) * C = A * (B * C)

2. (A + B) * C = A*C + B*C or A * (B + C) = A*B + A*C

# Advice 3 : How to calculate the determinant of order 4

Determinant (determinant) of the matrix is one of the most important concepts of linear algebra. The determinant of a matrix is a polynomial of square matrix elements. To calculate the determinant of the fourth order, you need to use the General rule for the computation of the determinant.

You will need

- Rule of the triangles

Instruction

1

A square matrix of the fourth order is a table of the numbers of four rows and four columns. Its determinant is calculated by the total recursive formula shown in Fig. M index is an additional minor of this matrix. Minor of a square matrix M of order n with index 1 at the top, and indices from 1 to n at the bottom, is the determinant of the matrix which is obtained from the source by striking out the first row and column j1...jn (j1...j4 columns in the case of a square matrix of the fourth order).

2

From this formula it follows that the resulting expression for the determinant of a square matrix of the fourth order will represent a sum of four terms. Each term will be the product of ((-1)^(1+j))aij, that is, one of the members of the first row of the matrix, taken with positive or negative sign, on a square matrix of the third order (the minor of a square matrix).

3

The minors, who represent a square matrix of the third order, we can already count on a well-known private formula, without the use of new minors. Determinants of square matrix of the third order can be calculated by the so-called "triangle rule." The formula for calculating the determinant in this case the output is not necessary, and you can remember its geometric scheme. This scheme is shown in the following figure. The result is |A| = a11*a22*a33+a12*a23*a31+a13*a21*a32-a11*a23*a32-a12*a21*a33-a13*a22*a31.

Consequently, the computed minors and the determinant of a square matrix of the fourth order can be calculated.

Consequently, the computed minors and the determinant of a square matrix of the fourth order can be calculated.