How to solve the formula Cramer
Cramer's rule is an algorithm to solve a system of linear equations using matrix. Method Author - Gabriel Cramer, who lived in the first half of the XVIII century.
Instruction how to solve by Cramer's rule
Let there be given a system of linear equations. It must be written in the matrix form. The basic matrix of the coefficients of the variables will be used. To record additional matrices will be needed and the free terms, usually located to the right of the equal sign.
Each of the variables must be a "serial number". For example, in all the equations of the system in the first place is x1, the second - x2, in the third - x3, etc. Then each of these variables will correspond to a column in the matrix.
To apply Cramer's method requires that the resulting matrix was square. This condition corresponds to equal the number of unknowns and the number of equations in the system.
Locate the main determinant of the matrix Δ. It should be zero: only in this case the solution will be unique and uniquely defined.
To record an additional determinant Δ (i), replace the i-th column is a column of free terms. The number of additional qualifiers will be equal to the number of variables in the system. Calculate all the determinants.
From these qualifiers you only find the value of the unknown. In general terms, the formula for finding the variable looks like: x (i) = Δ (i) / Δ.
Example. The system, consisting of three linear equations containing three unknowns x1, x2 and x3, has the form: a11 • x1 + a12 • x2 + a13 • x3 = b1, a21 • x1 + a22 • x2 + a23 • x3 = b2, a31 • x1 + a32 • x2 + a33 • x3 = b3.
From the coefficients of the unknown, record the main determinants: a11 a12 a13a21 a22 a23a31 a32 a33
Calculate it: Δ = a11 • a22 • a33 + a31 • a12 • a23 + a13 • a21 • a32 - a13 • a22 • a31 - a11 • a32 • a23 - a33 • a12 • a21.
Replacing the first column of free terms, make a first additional determinant: b1 a12 a13b2 a22 a23b3 a32 a33
A similar procedure to carry out the second and third columns: a11 b1 a13a21 b2 a23a31 b3 a33a11 a12 b1a21 a22 b2a31 a32 b3
Calculate the additional determinants: Δ (1) = b1 • a22 • a33 + b3 • a12 • a23 + a13 • b2 • a32 - a13 • a22 • b3 - b1 • a32 • a23 - a33 • a12 • b2.Δ (2) = a11 • b2 • a33 + a31 • b1 • a23 + a13 • a21 • b3 - a13 • b2 • a31 - a11 • b3 • a23 - a33 • b1 • a21.Δ (3) = a11 • a22 • b3 + a31 • a12 • b2 + b1 • a21 • a32 - b1 • a22 • a31 - a11 • a32 • b2 - b3 • a12 • a21.
Find unknown, write down the answer: x1 = Δ (1) / Δ, x2 = Δ (2) / Δ, x3 = Δ (3) / Δ.