One of tasks of the higher mathematics – the proof of compatibility of a system of the linear equations. The proof needs to be carried out according to the theorem of Kronkera-Kapelli according to which a system in common if the rank of its main matrix is equal to a rank of an expanded matrix.
Instruction
1. Write down the main matrix of a system. For this purpose bring the equations into a standard look (that is expose all coefficients in the same order if any of them is not present – write down, just with numerical coefficient "0"). Write out all coefficients in the form of the table, bracket it (the free members transferred to the right part do not consider).
2. In the same way write down an expanded matrix of a system, only in this case put vertical line on the right and write down a column of free members.
3. Count a rank of the main matrix, it is the greatest nonzero minor. The minor of the first order is any figure of a matrix, it is obvious that it is not equal to zero. To count a minor of the second order, take any two lines and any two columns (at you the table of four figures will turn out). Count determinant, increase the top left number by the lower right, subtract the work of the lower left and top right from the received number. At you the minor of the second order turned out.
4. It is more difficult to count a minor of the third order. For this purpose take any three lines and three columns, at you the table of nine numbers will turn out. Count determinant on a formula: ∆ = a11a22a33+ a12a23a31+ а21а32а13-а31а22а13-а12а21а33-а11а23а32 (the first figure of coefficient – the line number, the second figure – number of a column). You received a minor of the third order.
5. If in your system four or more equations, count also minors of the fourth (the fifth, etc.) orders. Choose the biggest, the minor not equal to zero is and there will be a rank of the main matrix.
6. In the same way find a rank of an expanded matrix. Pay attention if the quantity of the equations in your system coincides with a rank (for example, three equations, and a rank it is equal to 3) to count a rank of an expanded matrix there is no sense – obviously that it will also be equal to this number. In that case it is possible to draw safely a conclusion that the system of the linear equations in common.