Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last several lectures, we studied systems of two linear equations in two variables
and systems of three linear equations in three variables
and we have seen how these systems can be solved
using the methods of "substitution" or "elimination".
Although these methods can be used for systems with any number of variables or equations
as the number of variables and equations increases
the systems become much more difficult to solve.
Because of this, mathematicians have devised methods of solving systems of linear equations using matrices
along with a series of operations which can be performed on these matrices to solve a system
regardless of the number of equations and variables.
In this lecture, we will introduce matrices
and see how they can be used to represent systems of linear equations
in a compact, easy-to-manipulate form.
A matrix is a type of mathematical object
just as numbers
variables
and expressions are types of mathematical objects.
A matrix is a rectangular array containing individual "elements" or "entries"
enclosed in brackets.
The entries of a matrix can be numbers
symbols
expressions
or any type of mathematical object
for which the operations of addition and multiplication have been defined.
This requirement is necessary
since some matrix operations involve addition or multiplication of individual entries.
The entries of a matrix are arranged in rows
and columns
and a matrix can have any number of rows and any number of columns.
A matrix of "m" rows
and "n" columns
is said to be an "m by n" matrix.
For example, this matrix is a "four by five" matrix
so we say that the "dimension" or the "order" of this matrix is "four by five".
The order of a matrix is always given as the number of rows
followed by the number of columns.
The rows of a matrix are numbered from top to bottom, starting with row 1
and the columns are numbered from left to right starting with column 1.
The position of an entry in a matrix can be specified by its row and column numbers.
These are called the "indices" of the entry
and are written as subscripts
following a variable which represents the entries in the matrix.
For example, if we choose the variable "a" to represent an entry
and i and j as the subscripts
we can refer to the entries by writing "a"
where the first subscript, i is the "row index"
and the second subscript, j is the "column index" of a given entry.
For example, we can specify the entry in the third row
and second column of this matrix
as "a" with subscripts "3, 2".
In this array the value of that matrix entry is "1".
The "main diagonal" of a matrix is composed of all the entries
whose row and column numbers are the same.
One special type of matrix is a "square matrix".
Square matrices have an equal number of rows and columns.
If the number of rows and columns in a square matrix is "n"
we say that the square matrix is of "order n".
For example, this square matrix is of order 4.
An "upper triangular matrix" is a square matrix
whose entries below the main diagonal are all zero.
The entries in the main diagonal and above can have any value, zero or otherwise.
Likewise, a "lower triangular matrix" is a square matrix
where all the entries above the main diagonal are zero.
In a lower triangular matrix
the entries in the main diagonal and below can have any value.
A "diagonal matrix" is a square matrix
whose entries above and below the main diagonal are zero.
Other special type of matrices are matrices with only a single column
called "column matrices"
and matrices with a single row called "row matrices".
Another special type of matrix which is useful for solving systems of linear equations
is an "augmented matrix".
As we mentioned in the beginning of this lecture
augmented matrices are used to represent systems of linear equations
in a compact, easy-to-manipulate form.
Augmented matrices are like other matrices
with the addition of a vertical line
and an additional column of entries to the right of that line.
Each row of the augmented matrix is a shorthand representation of one equation of the system.
The columns to the left of the vertical line
contain the variable coefficients for each equation.
Any variable without a coefficient
is considered to have an implied coefficient of "1".
And any variable in the system which is missing from an equation
has an implied coefficient of "0".
The entries in the column to the right ofthe vertical line
correspond to the constants on the right side of each equation.
We can think of the vertical line in the augmented matrix
as corresponding to the equal signs in the equations.
Now let's look at a few examples of systems of linear equations
and see how to create their augmented matrix representations.
When creating the augmented matrix for a system of equations
remember to first make sure that the equations are in standard form
with the variable terms on the left side of the equal sign
and the constant terms on the right.
The variable terms should appear with the variables in alphabetical order
going from left to right.
And remember that any variables without coefficients
have an implied coefficient of "1".
Also remember that any variable in the system which is missing from an equation
has an implied coefficient of zero.
So for example, if this system of equations in three variables
contained any equations with less than three variables
then the equations might appear like this.
Before creating the augmented matrix
you may find it helpful to create a table of variable coefficients
placing the coefficients for each variable, including any missing variables
in the proper rows and columns of the table.
So let's start by creating an augmented matrix
for a system of two linear equations in two variables.
To create the augmented matrix for this system
we first collect all the coefficients and constants for each equation.
Since in the second equation, x has no coefficient, it's implied coefficient is "1".
We then collect the coefficients and constants for each equation
and enclose them in brackets
with a vertical line separating the coefficients from the constants.
To create an augmented matrix for this system of two linear equations in three variables
we start by filling in the implied coefficient of "1" for the x term in the second equation.
We then collect the coefficients and constants
and enclose them in brackets
with a vertical line separating the coefficients from the constants.
To create an augmented matrix for this system of three linear equations in three variables
we must fill in the implied coefficients of "1" for the terms with no coefficients
and the implied coefficient of "0" for the missing z term in the third equation.
We can then collect the coefficients and constants
and enclose them in brackets with a vertical line.
Likewise, to create an augmented matrix
for this system of four linear equations in four variables
we fill in the coefficients of "1"
and any missing terms
collect the coefficients and constants
and enclose them in brackets with a vertical line.
As a final example, we will create an augmented matrix
for this system of five linear equations in four variables.
We once again fill in the coefficients of "1"
and the missing terms
collect coefficients and constants
and create the augmented matrix.
So far, we have seen how to represent systems of linear equations using augmented matrices.
In the next lecture, we will introduce three matrix operations called "elementary row operations"
which are used to transform an augmented matrix
to a matrix representing an equivalent system of simpler equations
which indicate the solution set of the system.