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Dot Product

This article explores De Moivre's Theorem, proving its truth even for complex numbers with real and imaginary parts. We also simplify the understanding of determinants in matrices, incorporating the concepts of minors and cofactors. A detailed explanation of the relationship between the minor ( M_{ij} ) and the cofactor ( C_{ij} ) is provided. Additionally, the article covers Cramer’s Rule, a powerful method for solving systems of equations, showcasing the practical applications of these mathematical concepts.

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Dot Product

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  1. Dot Product

  2. Cross Product

  3. De Moivre’s Theorem DeMoivre's Theoremis true even if n is a complex number (has a real part and possibly an imaginary part), but when n is an integer we can prove the formula easily by using some basic trigonometry.

  4. Determinant of a Matrix For a matrix A defined as Where Cij is the cofactor and k is an integer between 1 and n

  5. Minor and Cofactor The minor, Mij, of the element a in a matrix A is the determinant of the matrix that remains after we delete the row i and column j containing aij. The relationship between cofactor, Cij, and minor, Mij, is defined as follows:

  6. Cramer’s Rule To solve a system of equations

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