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Algebra 2 [Lecture notes] by Ben Smith

By Ben Smith

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Extra resources for Algebra 2 [Lecture notes]

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Now it is your turn to try it. a. Balance equation F eCl2 + N a3 (P O4 ) −→ F e3 (P O4 )2 + N aCl by solving the system Cx = 0 where C is the coefficient matrix. by creating a 3×3 coefficient b. Balance the x1 Cu2 S +x2 O2 −→ x3 Cu+x4 SO2 ⎛ ⎞ x1 matrix C and solve the system Cx = b with ⎝ x2 ⎠ = det(C)C −1 b and x3 x4 = det(C). Project 7: Magic Squares A magic square is an arrangement of positive integers in a square grid, where the numbers in each row, in each column, and the numbers in the main diagonals, all add up to the same number.

C. Does A + B = A + B? d. B? Eigenvalues For each n × n matrix A, we can calculate the eigenvalues of A by finding the values for λ such that Ax = λx. The λ s are the eigenvalues for A and each eigenvalue has a corresponding eigenvector x. Another way to find the eigenvalues is to solve for λ in the characteristic equation |A − λI| = 0. Exercises: To calculate the eigenvalues of a matrix, Type Eigenvalues[The Name of the Matrix] 36 Exploring Linear Algebra 2 0 . Use your result to make a conjecture 0 3 about the values of eigenvalues of any diagonal matrix in general.

Encipher your message from part a. c. LHP d. Write up your findings and supporting mathematical argument. Project 3: Leontief Closed Production Model In a Closed Economy Leontief Model, each industry has a production level pi , and each industry i has a consumption level for product j, cij . If the economy is balanced, the total production of each industry must be equal to its total consumption and thus producing a linear system of equations. c11 p1 + c12 p2 + ... + c1k pk c21 p1 + c22 p2 + ... + c2k pk = p1 = p2 c31 p1 + c32 p2 + ...

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