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Linear Equations

      Let us consider n linear equations such that

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

      Let us consider:

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

      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:

image

      From here, we can easily get

image

      Solution: By expansion of determinants, we have:

image

      Which can be written as a1jA1j = a a1jA2j = 0 and a1jA3j = 0 [we know aijAij = a].

      Similarly, we have

image

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

image

      All nine of these equations can be combined into image.

      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, image and their matrices, image

image

      each of which is an n × n matrix.

      We shall now establish the following results:

      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 image. Hence, image.

image

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

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

      Proof: We have image, hence, image.

      Therefore,

image

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

image

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

image

      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:

image

      The above equation becomes

image

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

image

      for n unknown xi, where image

      image, where image is cofactor of image.

      image, which is called Cramer’s Rule,

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