Introduction to Matrix Operations
Matrices, the re presentation of a set of numbers in a uniform array, are used
engineering applications. From high end computer analysis packages to simple
bases, a matrix is a compact and useful tool in engineering analysis.
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 .
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 (2nd
column) is 5.
n Matlab, we place the row and column numbers in
parentheses . These numbers are
called indices into the matrix.
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
is that the two elements being summed must be of the same dimensions (same
of rows and columns.
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
rows in the second matrix. The resulting matrix has the same number of rows as
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 (R1, R2, and R3).
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
physical dimensions l and h are also known, l = 3 m and h = 0.5 m. The above
equations make up a system of three equations and three unknowns. These can be
in matrix form and we can let Matlab help up solve them.
Keeping in mind the definition of matrix multiplication, we can put this system
If we use symbols to represent the three matrices, you can represent this
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
matrix when multiplied by the original matrix. Keeping with the above equation,
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:
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
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
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
the previous 4 equations.
If Rint = 5 ohms, R1t= 100 ohms, Rmot = 50 ohms
and eint = 1.5 volts, solve this problem.
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