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Linear Equations

      Let us consider n linear equations such that

(1.10a)

      where x¹, x², …. xn are n unknown variables.

      Let us consider:

      For the expansion of det |ai j| in terms of cofactors we have

(1.10b)

      where a = |ai j| and the cofactor of ai j is Ai j.

      We can derive Cramer’s Rule for the solution of the system of n linear equations:

Now, multiplying both sides of (1.10a) by Ai j, we get

by (1.10b), we get, axj = biAi j.

      From here, we can easily get

Example 1.5.1. Show that , where a is a determinant ai jie a = |ai j| of order 3 and Ai j are cofactors of ai j.

      Solution: By expansion of determinants, we have:

      Which can be written as a_1jA^1j = a a_1jA^2j = 0 and a_1jA^3j = 0 [we know aijAij = a].

      Similarly, we have

      Using Kronecker Delta Notation, these can be combined into a single equation:

      All nine of these equations can be combined into .

1.6 Results on Matrices and Determinants of Systems

      It is known that if the range of the indices of a system of second order are from 1 to n, the number of components is n². Systems of second order are organized into three types: ai j, ai j, and their matrices,

      each of which is an n × n matrix.

      We shall now establish the following results:

Property 1.6.1. If , then and .

      Proof: We shall prove this result by taking the range of the indices from 1 to 2, but the results hold, in general, when they range from 1 to n.

      We get . Hence, .

      Taking the determinant of both sides, we get , as we know |AB| = |A||B|.

      Property 1.6.2. If , then, and , where (bik)^T is the transpose of

      Proof: We have , hence, .

      Therefore,

      Taking determinants of both sides, we get (since │AT│ = │A│).

Property 1.6.3. Let the cofactor of the element in the determinant be denoted by . Then, by summation convention we have

      If the cofactor of aij is represented by Akj, it is expressed by the equation:

      If we divide the cofactor Akj of the element of akj by the value a of the determinant, we form the normalized cofactor, represented by:

      The above equation becomes

      Property 1.6.4. Let us consider a system of n linear equations:

      for n unknown xi, where

      , where is cofactor of .

      , which is called Cramer’s Rule,

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