Critical Thinking Logical Proo Essay

1876 words - 8 pages

An argument given by professor Loren Johnson of the Mathematics Department of the University of California Santa Barbara states: The determinant of the n x n matrix A is the product of it's eigenvalues (Yaquib 303). In order to show if this argument is valid and sound we will need to define some essential terms. I am going to assume that a fair amount of Calculus is know to the reader in order to show whether or not this argument is valid and sound. Matrices are used in linear algebra to discuss systems of equations. The matrix itself is composed of the terms preceding each of the variables in each equation of the system. An example of a system of three equations would be: 2x + 3y + 4z, x +3y and 6x + 2y + 2z. The first row of the matrix for this system is [2 1 6], the second would be [3 3 2] and the third would be [4 0 2] which we will call A. Using this information we can define the determinant as being the sum of all possible elementary signed products from A. This can only be achieved if A is an n x n matrix, where n represent the number of rows and columns. The signed elementary products of A can be defined as 1 when the permutation of the elementary products is even and -1 when the permutation of the elementary products is odd (Hughes-Hallet 20). These two numbers are then multiplied by their respective permutation and the whole lot is added together. When all calculations are said and done this results in a number for matrix A, in this case -58. This now leads us to the definition of an eigenvalue. Since we have already defined A as a n x n matrix we can say that the eigenvalues are the solutions of the equation obtained from taking the determinant of the matrix A subtracted from the product of the identity matrix and the variable lambda (Hughes-Hallet 90). The identity matrix is defined as a matrix with ones along it's diagonal and zeros everywhere else (Yaquib 23). Now having a rough idea of what the determinant and eigenvalues are we can proceed with discussion of the argument.Is this a valid argument? The definition of validity explains in order to be valid the argument must have the characteristic that it's conclusion absolutely follows from the premises. We can rewrite this argument into a more standard argumentative form; If A is an n x n matrix then the determinant of A is the product of it?s eigenvalues. The antecedent of this argument can be represented as P and the consequent of this argument can be represented as Q. This means that the argument can now be represented in symbolic logic as P-Q or if P then Q. We know from our extensive knowledge of truth tables that the only time a conditional statement is false is when the antecedent is true and the consequent is false. Therefore we only need to worry about the consequent since the antecedent is what we are going to assume to be true in our argument. Since we are assuming that A is any n x n matrix then it can follow from the definition of determinants that A has a determinant....

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