Introduction to Matrix Operations

Matrices, the re presentation of a set of numbers in a uniform array, are used
widely in

engineering applications. From high end computer analysis packages to simple
data

bases, a matrix is a compact and useful tool in engineering analysis.

In this handout, we briefly cover the rules of fundamental
matrix operations and a few

examples of how they are used for solving systems of equations.

__Matrix representation__

A matrix can be a large array of numbers denoted by a
single symbol. The symbol used to

represent the matrix is usually in bold font, or alternately, under lined .

For example:

R is called a 2 x 3 matrix (2 rows by 3 columns). In
Matlab, we would enter the matrix

into the workspace using the fol lowing command .

>> R = [3 8 10;-4 6 5];

We can refer to a single element of the matrix by
specifying the elements position in the

matrix, it's row and column number. For example, in R, the 2,3 element (2^{nd}
row, 3^{rd}

column) is 5.

n Matlab, we place the row and column numbers in
parentheses . These numbers are

called indices into the matrix.

>> R(2,3)

would return the 2,3 element, which is 5.

__Matrix Addition and Subtraction.__

Addition and subtraction of matrices is straightforward.
The sum of two matrix is the

result of summing the element individually. AN important implication of this
definition

is that the two elements being summed must be of the same dimensions (same
numbers

of rows and columns.

Let:

Then, if C = A + B,

__Matrix Multiplication__

Multiplication of two matrices is more complicated.
Essentially the rows of the first

matrix are combined with the columns of the second matrix in a systematic
fashion . The

expression below shows the general rule.

where

Dimensional consistency is an important aspect of matrix
multiplication. The above rule

implies that the number of columns in the first matrix must be equal to the
number of

rows in the second matrix. The resulting matrix has the same number of rows as
the first

and the same number of columns as the second. When MATLAB generates and error

message that "inner dimensions must agree" it has found a case in which you are

attempting to violate this rule.

__Systems of equations and matrix multiplication__

To get a better understanding of these issues, consider
the following two problems, one

taken from sophomore mechanical and civil engineering classes (ENGR 210) and one

taken from sophomore electrical engineering classes (ENGR 240).

**Problem 1: Equilibrium Mechanics.**

The figure below shows a pickup truck undergoing
acceleration. We know the mass of

the pickup truck, the location of the center of gravity and the magnitude of the

acceleration. Find the force experienced at the tires (R_{1}, R_{2}, and R_{3}).

From the principles of equilibrium and Newton second law
of motion , we can write the

following three equations involving the unknown forces on the tires:

The mass of the pickup is approximately 1000 kg and it is accelerating at 3
m/s/s. The

physical dimensions l and h are also known, l = 3 m and h = 0.5 m. The above
three

equations make up a system of three equations and three unknowns. These can be
placed

in matrix form and we can let Matlab help up solve them.

First, we must put the equations in matrix form. Start by
realizing that all three equations

can be seen to involve all three unknowns, just some of them have coefficients
of zero .

Keeping in mind the definition of matrix multiplication, we can put this system
in matrix

form:

If we use symbols to represent the three matrices, you can represent this
equation as

shown below:

**A R = B**

We solve this equation (that is, find values for the
forces in R), by the process of matrix

inversion. The inverse of a given matrix is that matrix, which will yield the
identity

matrix when multiplied by the original matrix. Keeping with the above equation,
we can

multiply both sides by the identity matrix:

**A**^{-1} A R = A^{-1} B

Which simplifies to:

**R = A**^{-1} B

Matlab makes this easy. Given the values that were give
for this problem, we can easily

enter the matrices:

>> A=[1 1 0;0 0 1;1.5 ?.5 0.5]

>> B=[1000*9.81;1000*3;0]

>> R= inv(A)*B

It is left to the student to compute the final answers and
verify their correctness.

**Problem 2: Multiple Loop Circuits**

The sketch below shows a circuit that can be used to model
a battery powering two loads

in parallel, say a motor and a light bulb. The chemical reaction inside the
battery provides

an electrical potential of about 1.5 volts but internal losses make the actual
voltage on the

battery terminals somewhat less. These losses are modeled by an internal
resistance.

Using basic principles of circuit analysis, we can easily
right the following 4 equations

summarizing the circuit behavior.

It is left to the student to verify that the matrix form below is a proper
representation of

the previous 4 equations.

If R_{int} = 5 ohms, R_{1t}= 100 ohms, R_{mot} = 50 ohms
and e_{int} = 1.5 volts, solve this problem.