How to calculate determinant of the second order

How to calculate determinant of the second order

Determinant – one of concepts of matrix algebra. It is the square matrix consisting of four elements and to calculate determinant of the second order, it is necessary to use a decomposition formula on the first line.

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Instruction

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The determinant of a square matrix is a number which is used in various calculations. It is irreplaceable during the finding of the return matrix, minors, algebraic additions, operation of division of matrixes, but most often need of transition to determinant arises at the decision of systems of the linear equations.

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To calculate determinant of the second order, it is necessary to use a decomposition formula on the first line. It is equal to a difference of paired works of the elements of a matrix located on the main and collateral diagonal respectively:

? = a11 • a22 – a12 • a21.

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The matrix of the second order represents set of four elements located on two lines and columns. These numbers correspond to coefficients of system of the equations with two unknown who are applied by consideration of a set of applied problems, for example, of the economic.

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Transition to compact matrix calculations helps to define two things quickly: first, whether this system has the decision, secondly, to find it. A sufficient condition of existence of the decision is the determinant inequality to zero. It is connected with that at calculation of unknown components of the equations this number costs in a denominator.

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So, let there is a system from two equations with two variables x and y. Each equation consists of couple of coefficients and the free member. Then three matrixes of the second order are formed: elements of the first – coefficients at x and y, the second contains free members instead of coefficients at x, and the third – instead of numerical multipliers at a variable y.

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Then values of unknown can be calculated as follows:

x =? x/?; y =? y/?.

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After expression through the corresponding elements of matrixes, it turns out:

? = a1 • b2 – b2 • a1;
? x = c1 • b2 – b1 • c2> x = (c1 • b2 – b1 • c2) / (a1 • b2 – b2 • a1);
? y = a1 • c2 – c1 • a2> y = (a1 • c2 – c1 • a2) / (a1 • b2 – b2 • a1).