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Properties of Determinants

A matrix is an array of numbers and each square matrix is made up of numbers called determinants.
The determinant is used in the formula method for solving a system of equations which is called Cramer's rule. The strength of Cramer's Rule is that it can be applied to a linear system of n equations with n unknows.
The determinant of a n-by-n matrix can be calculated as a sum of determinants of submatrices.
Cramer's rule for system 3x3:
Consider now the 3 x 3 system

a11x1 + a12x2 + a13x3 = b1
a21x1 + a22x2 + a23x3 = b2
a31x1 + a32x2 + a33x3 = b3

Solution of the system of equations in three unknowns is given by the formulas

x1 = det(D1)/det(D)
x2 = det(D2)/det(D)
x3 = det(D3)/det(D)

where det(D1) is the determinant of matrix D1
b1 a12 a13
b2 a22 a23
b3 a22 a23
 det(D2) is the determinant of matrix D2
a11 b1 a13
a21 b2 a23
a31 b3 a33
det(D3) is the determinant of matrix D3
a11 a12 b1
a21 a22 b2
a31 a32 b3
det(D) is the determinant of matrix D
a11 a12 a13
a21 a22 a23
a31 a32 a33
Cramer's rule holds for n unknowns
xi = det(Di)/det(D), i=1,2,..,n

where matrices Di are coefficient matrices where coefficients of xi are replaced by the constants bi.
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