CHEM%20140L%20Session%2006

CHEM%20140L%20Session%2006 - Session 06 Algebra and...

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6-1 Session 06 Algebra and Calculus in Mathcad Topics In this session the following spreadsheet concepts will be explored: 1. A Review of Linear Algebra Determinants Matrices 2. Working with Matrices in Mathcad C Defining a Matrix C Modifying Matrices C Matrix Math Operations 3. Working with Determinants in Mathcad 4. A Review of Differentiation and Integration C Differentiation C Integration 5. Numerical Differentiation in Mathcad 6. Numerical Integration in Mathcad
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CHEM 140L Session 6 - Algebra and Calculus in Mathcad 6-2 A Review of Linear Algebra = Determinants A determinant is a square array of numbers which represents a certain sum of products. Determinants are written within a pair of vertical lines | |. Shown below is an example of a 3 × 3 determinant, with 3 rows and 3 columns. 90 4 35 1 402 −− The final result of the determinant, after multiplying and simplifying the elements of a determinant is a single number (i.e. a scalar quantity). = Calculating a 2 × 2 Determinant In general, to find the value of a 2 × 2 determinant with the elements a , b , c , and d , we multiply the diagonals (top left × bottom right first), then subtract: ab cd ad bc =⋅−⋅ Consider the following example: () 41 23 43 12 12 2 10 × =− = The final result of this determinant is 10, a single number.
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CHEM 140L Session 6 - Algebra and Calculus in Mathcad 6-3 = Calculating a 3 × 3 Determinant Consider the following general 3 × 3 determinant: abc 111 222 333 We can evaluate a 3 × 3 determinant by using the method of expansion by minors , otherwise known as the Laplace expansion of a determinant. The expansion by minors requires the definition of a cofactor : the 2 × 2 determinant bc 22 33 is called the cofactor of a 1 , for the above 3 × 3 determinant. The cofactor is from the elements that are not in the same row as a 1 and not in the same column as a 1 . Similarly, the determinant 11 is the cofactor of a 2 . It is formed from the elements not in the same row as a 2 and not in the same column a 2 . To evaluate the 3 × 3 determinant, we would take the elements of the top row and multiply by the determinant of corresponding cofactors. The sign alternates between
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CHEM 140L Session 6 - Algebra and Calculus in Mathcad 6-4 products. In the case of a 3 × 3 determinant, the middle product is subtracted and the final product is added. () abc a bc b ac c ab ab c b c ba c a c ca b a b 111 222 333 1 22 33 1 1 12 3 3 2 1 2 3 3 2 3 3 2 =−+ × × × × Larger determinants can be evaluated by going across the first row, by determining the cofactor for each element in the first row and then multiplying each first row element by its corresponding cofactor. The signs between products alternate, such that the first product term is positive, the second product term is negative, the third product term is positive, the fourth product term is negative, and so on.
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CHEM%20140L%20Session%2006 - Session 06 Algebra and...

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