Further Analytical Methods Essay Example

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Lecturer name

Topic: Assignment 4 – Matrices and Vector Geometry

Task (LO 3.1)

Using Kirchhoff’s circuit laws, find the current flowing in each branch of this circuit by first setting up 33 simultaneous equations and then using the Gaussian Elimination method to solve the set of equations.

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Hint : Apply Kirchhoff’s voltage law to the loops with currents I1, I2 and I3. You could check your solution using Kirchhoff’s current law (also known as Kirchhoff’s junction rule) at the four nodes.

This has 6 branches and 4 nodes

L= 7-(4-1)=4

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Solution

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Task (LO 3.2)

For the given set of equations below (as determined by your data set number) solve the equations TWICE:

  1. firstly, using gaussian elimination in augmented matrix format;

  2. secondly, using inverse matrices.

A submission in Excel for LO 3.2.2 is preferred.

Systems of Equations (Data sets 1 to 12)

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Solution

Add 1/5 x row 1 to row 2

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Less 2/5 row 1 to row 3

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Add 3x row 2 to 7xrow 3

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Less 2 times row 1 from row 2

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Add 7 times row 1 to row 3

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Swap 2 and 3

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Less 2 times row 1 from row 3

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Add 1/2 times row 3 to row 1

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Add 1/7 times row 2 to row 3

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Add 35/65 times row 3 to row 1

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Less 16 times row 3 from row 3

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add 2 times row 1 to row 3

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Divide by row 1 by -1

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less 1/5 times row 5 to row 1

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Less 9 times row 1 from row 2

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Divide by 2 by -4

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Less 5 times row 2 to row 3

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Divide -46.75 to row 3

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Less 2.88 row 3 from row 1

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Less 7.55 row 3 from row 2

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Less 9x row1 from row 3 and add 5x row1 to row 2

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add 1/8x row3 to row 1 and less1/2 of row3 from row 2

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add 8 x row2 to row 3 and

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add 1/7 x row3 to row 1 and 2/23 of row 3 to row 2

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Add 7x row1 to row 2 and add 9x row1 to row 3

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less 1/2x row3 to row 1 and less 1/6x row2 from row 3

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add 8x row3 to row 1 and less 15x row3 from row 2

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Add row 3 to row 2 and less 5 x row 1 from row 3

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Add 3xrow 1 to row 2 and add 11 x row 1 from row 3

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Less 1/2xrow 2 from row 1 and add 3 x row 3from row 2

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less 8 x row 1 from row 3 and less 3 x row 1 from room 2

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Less 2 x row 2 from row 3 and add1/12 x row 2 to room 1

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Less 1/3 x row 3 from row 1 and less row 3 to room 2

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Add 2 times row 3 to row 1 and less 9 times row 3 from row 2

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Less row 1 from to row 3

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Less 3/2 times row 3 from to row 1 and Less 1/4 times row 2 from to row 3

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Less row 3 from to row 1 and add 7 times row 3 to row 2

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Less row 2 from row 1

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Less rows 2 from row 1

There is no visible solution to the problem

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Divide row 1 by 6 and row 3 by 2

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Add 4times row 1 to row 3 and Add 9 times row 1 to row 2

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Less 1/8times row 3 from row 1 and less 8times row 2 from row 3

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Less 17times row 3 from row 2 and less 1.125times row 3 from row 1

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Divide row 1 by 3

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Add 6 times row 1 to row 3

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Less ½ row 3 to row 1

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add 9 times row 1 to row 2

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Divide row 2 by -8 times row 1 to row 2

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Divide row 2 by -8 times row 1 to row 2

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Add 2 times row 2 to row 3 the divide row 3 by -15.875

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Add 3.4375 times row 3 to row 2 and less 3.5times row 3 to row 1.

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Less 2 times row 3 from row 1

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add 8 times row 1 from row 2

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less row 1 from row 3

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Add 1/3 row 3 to row 1

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Add 4 times row 3 to row 2

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less 3 times row 2 to row 3

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Add 10 times row 3 to row 1

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add 75 times row 3 to row 2

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Task (LO 3.3)

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In the diagram above, use the dot product to find the minimum angle that the length marked A makes with the left hand wall. x is your dataset number.

a · b = |a| × |b| × cos(θ)

11+x · 2 = (4) × (2) × cos(k)