Introduction

DEFINITION: A matrix is defined as an ordered rectangular array of numbers. They

can be used to represent systems of linear equations, as will be explained below.

Here are a couple of examples of different types of matrices:

Symmetri Diagonal Upper Lower Zero Identit
c Triangula Triangula y
1 2 3 1 0 0 12 3 100 000 100
2 0 -5 0 4 0 0 7 -5 -4 7 0 000 010
3 -5 6 0 0 6 0 0 -4 12 5 3 000 001

And a fully expanded mon matrix A, would look like this:

A=

or in a more compact form: 4-(4)

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Matrix Addition and Subtraction
DEFINITION: Two matrices A and 13 can be added or subtracted if and only if their dimensions
are the same (ie both matrices have the same mumber of rows and columus. Take

Addition
If A and B above are nutrices of the same type then the sum is found by adding the
corresponding elements ab

Here is an example of adding A and B together.

Subtraction
If A and B are matrices of the same type then the subtraction is found by subtracting the
corresponding elements.a-bur
Here is an example of subtracting matrices.

Matrix Multiplication
DEFINITION: When the number of columns of the first matrix is

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The same as the number of rows in the second matrix then matrix multiplication can be
performed.
Here is an example of matrix multiplication for two 2×2 matrices.

Here is an example of matrix multiplication for two 3×3 matrices.

Now lets look at the nxn matrix case, Where A has dimensions men, B. has dimensions np. Then
the product of A and B is the matrix C, which has dimensions map. The ith element of matrix C
is found by multiplying the entries of the “ row of A with the corresponding entries th in the
column of B and summing the n terms. The elements of C are:

Note: That A B is not the same as B⭑Note: That A B BXA

Transpose of Matrices
DEFINITION: The transpose of a matrix is found by exchanging

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Rows for columns i.e. Matrix A = (a) and the transpose of A is: A-(a) where j is the column
number and I is the row number of
Matrix A.
For example, the transpose of a matrix would be:

In the case of a square matrix (m=n), the transpose can be used to check if a matrix is
symmetric. For a symmetric matrix A = A

The Determinant of a Matrix

DEFINITION: Determinants play an important role in finding the

Inverse of a matrix and also in solving systems of linear equations. In the following we assume
we have a square matrix (m=n). The determinant of a matrix A will be denoted by det(A) or [A].
Firstly the determinant of a 2×2 and 3×3 matrix will be introduced, then the non case will be
shown.

Determinant of a 2×2 matrix

Assuming A is an arbitrary 2×2 matrix A, where the elements are given by:

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Then the determinant of a this matrix is as follows:

Determinant of a 3×3 matrix

The determinant of a 3×3 matrix is a little more tricky and is found as follows (for this case
assume A is an arbitrary 3×3 matrix A, where the elements are given below).

Then the determinant of a this matrix is as follows:

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Determinant of a non matrix
For the general case, where A is an non matrix the determinant is given by:

Where the coefficients a are given by the relation:

Where B, is the determinant of the (n-1) (n-1) matrix that is obtained by deleting row I and
column j. This coefficient a, is also called the cofactor of a

The Inverse of a Matrix

DEFINITION: Assuming we have a square matrix A, which is
Non-singular (i.e. det(A) does not equal zero), then there exists an n-n matrix A which is called
the inverse of A, such that this property holds:

AAAA=1, where I is the identity matrix.

The inverse of a 2×2 matrix

Take for example a arbitury 2×2 Matrix A whose determinant (ad-bc) is not equal to zero.

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The inverse of a non matrix
The inverse of a general non matrix A can be found by using the following equation.
Deti Al

Where the adj(A) denotes the adjoint (or adjugate) of a matrix. It can be calculated by the
following method:

. Given the nxn matrix A, define B = b

To be the matrix whose coefficients are found by taking the determinant of the (n-1) x (n-1)
matrix obtained by deleting the I” row and j” column of A. The terms of B (i.e. B = b) are known
as the cofactors of A.

Define the matrix C, where

The transpose of C (ie. C) is called the adjoint of matrix A.

Lastly to find the inverse of A divide the matrix C by the determinant of A to give its inverse

Transpose of Matrices

DEFINITION: The transpose of a matrix is found by exchanging

Rows for columns i.e. Matrix A = (a) and the transpose of A is: A-(a) where j is the column
number and I is the row number of

Matrix A.

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For example, the transpose of a matrix would be:

In the case of a square matrix (m=n), the transpose can be used to check if a matrix is
symmetric. For a symmetric matrix A = A

The Determinant of a Matrix

DEFINITION: Determinants play an important role in finding the

Inverse of a matrix and also in solving systems of linear equations. In the following we assume
we have a square matrix (m=n). The determinant of a matrix A will be denoted by det(A) or [A].
Firstly the determinant of a 2×2 and 3×3 matrix will be introduced, then the non case will be
shown.

A matrix Is sort of like a “box” of information where you are keeping track of things both right
and left (columns), and up and down (rows). Usually a matrix contains numbers or algebraic
expressions. You may have heard matrices called arrays, especially in computer science.

As an example, if you had three sisters, and you wanted an easy way to store their age and
number of pairs of shoes, you could store this information in a matrix. The actual matrix is
inside and includes the brackets:

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Step-by-step explanation:

For instance, you might store this information in a matrix if you had three sisters and needed a
simple way to keep track of their ages and the number of pairs of shoes they owned. Matrix
representations of linear mappings without additional details enable explicit computations in
linear algebra. As a result, a significant portion of linear algebra involves the study of matrices,
and the majority of the characteristics and operations of abstract linear algebra may be
described in terms of matrices. The composition of linear maps, for instance, is represented by
matrix multiplication.

Not every matrix has a connection to linear algebra. This is especially true for incidence
matrices and adjacency matrices in graph theory. Since the focus of this page is on matrices
used in linear algebra, all matrices used here represent linear maps unless otherwise stated.

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Ex.1
Now, we talked about identifying the x above. We figured out that we must call one of the
unknowns x and the other 10-x. So now, let’s plug that in.

.10(x)+.30(10-x) = .15(10).
Now, let’s solve it!
Step 1: Foil
.10x+3-.3x= 1.5
Step 2: X’s on one side by subtracting 3 from both sides
.1x-.3x= -1.5
Step 3: Combine like terms-
.2x= -1.5
Step 4: Divide by –
.2X= 7.5
Step 5: Find the other unknown
X=7.5
10-x= 10-7.5= 2.5
10 (7.5) + .30(2.5) = .15(10)
75+.75= 1.5

Ex.2
A tank contains 20 gallons of a mixture of alcohol which is 40% alcohol by volume. How much of
the mixture should be removed and replaced by an equal volume of water so that the resulting
solution will 25% alcohol by volume?

40[alcohol]/.25= 20/20 -x[resulting expression common to mixture problems]. So, by cross

multiplying, the equation becomes:
40(20 – x) = .25(20) | 8 – .40x = 5 | 40 – 2x = 25 | -2x = -15 | Answer: x = 7.5 gallons

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