Linear algebra, matlab knowledge required, handshake required

Drexel University, College of Engineering 2015-2016 Academic Year

Drexel University Office of the Dean of the College of Engineering ENGR 231 – Linear Engineering Systems

Homework 4 – Week 4

Textbook Reading Assignment – Schaum’s Outlines – Linear Algebra 5th Edition

· Section 3.9, 3.10

· Section 8,1, 8.3, 8.5, 8.6, 8.7, 8.8

· Note: Not all topics in these sections were covered. Please read portions of these sections as necessary to understand the material taught in lecture.

Exercise 1: A system of linear equations can be represented in 4 equivalent ways as shown in lecture

As a system of n-equations As a matrix equation

As an augmented matrix As a vector equation

For each of the following systems, one of these forms is given. Complete each table by giving the three missing alternative representations. Be able to identify the number of unknown variables, and whether or not the given system can have a unique solution without solving the equation.

a. Complete this table of equivalent representations.

System of Equations	The Matrix Equation
The Augmented Matrix	The Vector Equation:

b. Complete this table of equivalent representations.

System of Equations	The Matrix Equation
The Augmented Matrix	The Vector Equation:

c. Complete this table of equivalent representations.

System of Equations	The Matrix Equation
The Augmented Matrix Assume the unknown variables are .	The Vector Equation:

d. Complete this table of equivalent representations.

System of Equations	The Matrix Equation
The Augmented Matrix	The Vector Equation:

Exercise 2: Find the determinant for each of the following matrices.

a. Use the arrow rule, the 2x2 formula or any matrix property to find these simple determinants.

The determinant of is: …………

The determinant of is: …………

The determinant of is: …………

The determinant of is: …………

b. Use cofactor expansions or any matrix property to find these determinants.

The determinant of is:

…………

The determinant of is:

…………

The determinant of is:

…………

The determinant of is:

…………

The determinant of is:

…………

The determinant of is:

…………

Exercise 3: In the following, use the matrices defined below. Calculate the quantities shown for each operation, and explain any cases that cannot be computed. If you need to remind yourself how to multiply matrices, please check the text.

Here, T represents the matrix transpose operator, and is the identity matrix.

Continue the rest here, finding all 16 combinations.

Exercise 4: Cramer’s Rule

a. Solve the following system of equations using Cramer’s Rule.

First write out the matrix A and the vector .

Form the new matrix by replacing column 1 of A with . (Hey! It's the same!)

Form the new matrix by replacing column 2 of A with .

Find each unknown as a simple fraction: - - -

- - -

b. Solve this system of equations using Cramer’s Rule.

Use steps similar to those shown above.

c. Solve this system of equations using Cramer’s Rule.

Exercise 5: MATLAB - MATLAB and concepts related to the laboratory exercises in week 4.

Write the MATLAB code to perform the following tasks:

a. Enter the following matrix A and vector in the MATLAB environment:

b. Write an expression to obtain the first column of A.

c. Find the determinant of A.

d. Is Cramer's rule applicable for this matrix?

If it is, find all the unknowns using Cramer's simple formula:

Exercise 6:

MATLAB

a. Write a program to find the determinant of a 2x2 matrix using the formula given in class.

Verify your function works by checking it with the det function from MATLAB.

b. Write a program to find the determinant of a 3x3 matrix using the formula given in class.

Verify your function works by checking it with the det function from MATLAB.

Exercise 7: solving the problem in the picture

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