Mathematical Excursion into Matrices and Determinants

A matrix can be defined as a two-dimensional arrangement of elements (numbers, variables, vectors, etc.) set up in rows and columns. The elements a are indexed as follows  

The dimension of a matrix is defined by the number of rows (m) and columns ( and is called an k X m matrix. 

A square matrix has the eigenvalue A if there is a vector x fulfilling the equation Ax = Ax. The result of this equation is that indefinite numbers of vectors could be multiplied with any constants. Anyway, to calculate the eigenvalues and the eigenvectors of a matrix, the characteristic polynomial can be used. Therefore (A - AE)x = 0 characterizes the determinant (A - AE) with the identity matrix E (i.e., the X matrix). Solutions can be obtained when this determinant is set to zero. 

The inside terms of the determinant represent the characteristic polynomial P(A) of A  

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