Instruction

1

*To compute determinant (Det A) of a matrix of dimension 5x5 swipe the decomposition of the elements in the first row. To do this, take the first element of this string and remove from the matrix the row and column at the intersection where it is located. Write down the formula of the first element and the determinant of the resulting matrix 4 order: a11*detM1 – this will be the first term to find Det A. the remaining four-digit matrix M1 you will need later to find the determinant of the (more minor).*

2

*Similarly, consistently cross out the column and row that contains 2, 3, 4 and 5 element of the first row of the initial matrix, and find for each one the corresponding 4x4 matrix. Write the product of these elements on additional minors: a12*detM2, a13*detM3, a14*detM4, a15*detM5.*

3

Find the determinants of the resulting matrices of order 4. To do this, swipe again in the same way the reduction of the dimensionality. The first element b11 of the matrix M1 multiply by the determinant of the remaining 3x3 matrix (C1). The same three-dimensional determinant of a matrix is easily calculated by the formula: detC1 = c11* c22*c33 + c13* c21*c32 + c12* c23*c31 - c21* c12*c33 - c13* c22*c31 - c11* c32*c23, where cij is the elements of the resulting matrix C1.

4

Next, consider similarly the second element b12 for the matrix M1 and compute its product with the corresponding additional minor detC2 obtained three-dimensional matrix. In the same way find works for 3 and 4 elements of first matrix of order 4. Then determine the required additional minor of the matrix detМ1. For this, according to the formula of decomposition by row, write down the expression: detМ1 = b11*detC1 - b12*detC2 + b13*detC3 - b14*detC4. You got the first term, required to find Det A.

5

Calculate the other terms of the determinant of the matrix of the fifth order, similarly lowering the dimension of each matrix of order 4. The final formula is: Det A = a11*detM1 - a12*detM2 + a13*detM3 - a14*detM4 + a15*detM5.

# Advice 2 : 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.

# Advice 3 : How to calculate matrix in excel

To calculate the values of the matrix, or perform other mathematical calculations using Microsoft Office Excel. You can also use free and its analogues, the principle of operation here will be practically the same.

You will need

- - Microsoft Office Excel.

Instruction

1

Start Microsoft Office Excel. In the data input screen you fill in this

**matrix**for subsequent computation of its determinant. Highlight one of the unoccupied cells in the table, enter the following formula: “=MODIED(ak:fg)”. In this case, ak will denote the coordinates of the upper left corner of the matrix, and fg is the bottom right. To obtain the determinant, press Enter. The desired value will be displayed in the chosen empty cell.2

Use the Excel functionality to calculate and other values. In case you do not know how to use formulas in Microsoft Office Excel, download special themed literature, and after reading you will be quite easy to navigate in this program.

3

Carefully read the names of the values of formulas in this software, because improper input you can spoil all the results, especially those who perform several of the same calculations for one formula at a time.

4

From time to time test received in Microsoft Office Excel the results of a calculation. This is due to the fact that the system could occur any change with time, in particular this applies to those who performs work on the template. Always it is useful once again to compare the results from several ongoing calculations.

5

Also, when working with formulas-be very careful and avoid your computer viruses. Even in case of operation with formulas in Microsoft Office Excel you need a one-time, review the functionality of this program to a greater extent, because these skills will help you in the future to better understand the automation of accounting and use Excel to perform certain tasks.

# Advice 4 : How to find the determinant of a matrix of order 3

**Matrices**exist for displaying and solving systems of linear equations. One of the steps in the algorithm of finding a solution is finding the determinant or determinant. The matrix is 3 orders of magnitude is a square matrix of dimension 3x3.

Instruction

1

The diagonal from the upper left element to bottom right called the main diagonal of a square matrix. From the right upper element to the lower-left – side. The matrix is 3 orders of magnitude has the form:a11 a12 a22 a13a21 a23a31 a32 a33

2

To find the determinant of the matrix of the third order there is a clear algorithm. First, summarize the main diagonal elements: a11+a22+a33. Then, the lower left element of the a31 with moderate elements of the first row and third column: a31+a12+a23 (visually turns the triangle). Another triangle – top-right element a13 and the median of the elements of the third row and first column: a13+a21+a32. All the data components go into the determinant with the plus sign.

3

Now you can go to term with the sign "minus". First, it's a side diagonal: a13+a22+a31. Second, the two triangles: a11+a23+a32 and a33+a12+a21. The ultimate formula for finding the determinant is: Δ=a11+a22+a33+a31+a12+a23+a13+a21+a32-(a13+a22+a31)-(a11+a23+a32)-(a33+a12+a21). The formula is pretty bulky, but after some time of practice it becomes familiar and "work" on the machine.

4

In some cases, it is easy to see that

**the determinant**of the matrix equal to zero. The determinant is zero if any two rows or two columns the same, is proportional to or linearly dependent. If at least one row or one column consists entirely of zeros,**the determinant of**the entire matrix equals zero.5

Sometimes to find

**the determinant**of a matrix is more convenient and easier to use matrix transformations: algebraic addition of rows and columns among themselves, making the total multiplier of the row (column) for the sign of the determinant, domlounge all elements of a row or column to the same number. For the transformation matrices is important to know their basic properties.Useful advice

For the calculation of the determinant there are many specific methods, but, as a rule, in the case of matrices of the third order to apply them impractical.