What is Matrix algebra?
 Matrix algebra is a means of making calculations upon arrays of numbers (or data).
 Most data sets are matrixtype
Why use it?
 Matrix algebra makes mathematical expression and computation easier.
 It allows you to get rid of cumbersome notation, concentrate on the concepts involved and understand where your results come from.
Definitions
 Matrices – definitions
 Matrix notation
 Equal matrices
 Addition and subtraction of matrices
 Scalar multiplication of matrices
 Transpose of a matrix
 Special matrices
 Multiplication of two or more matrices
 Determinant of a square matrix
 Inverse of a square matrix
 Solution of a set of linear equations
 Eigenvalues and eigenvectors
Matrices – definitions

A matrixis a set of real or complex numbers (called elements) arranged in rows and columns to form a rectangular array.

A matrix having m rows and n columns is called an m × n matrix.
For example:
$\left(\begin{array}{ccc}5& 7& 2\\ 6& 3& 8\end{array}\right)$
is a 2 × 3 matrix.

A matrix is a rectangular array of numbers or symbolic elements
In many applications, the rows of a matrix will represent individuals cases (people, items, plants, animals,...) and columns will represent attributes or characteristics
The dimension of a matrix is its number of rows and columns, often denoted as r x c (r rows by c columns)
Can be represented in full form or abbreviated form:

scalar a scalar is a number (denoted with regular type: 1 or 22).

Vector A matrix with one column (column vector) or one row (row vector).

Row vector
A row vector consists of a single row. For example:
$\left(\begin{array}{ccc}5& 7& 2\\ 6& 3& 8\end{array}\right)$

Column vector
A column matrix consists of a single column. For example:
$\left(\begin{array}{c}6\\ 3\\ 8\end{array}\right)$

Double suffix notation
Each element of a matrix has its own address denoted by double suffices, the first indicating the row and the second indicating the column. For example, the elements of 3 × 4 matrix can be written as:
Matrix Notation
Where there is no ambiguity a matrix can be represented by a single general element in brackets or by a capital letter in bold type.
Equal matrices
Two matrices are equal if corresponding elements throughout are equal
Addition and Subtraction
Where
Addition and subtraction of matrices
Two matrices are added (or subtracted) by adding (or subtracting) corresponding elements. For example:
Multiplication of matrices
 Scalar multiplication
 Multiplication of two or more matrices
Scalar multiplication
To multiply a matrix by a single number (a scalar), each individual element of the matrix is multiplied by that number. For example;
That is:
Properties of matrix addition and scalar multiplication
Then :

$A+BA=ABA+AA$
Commutative law for addition

$A+(B+C)=(A+B)+C$
Associative law for addition

$\left(cd\right)A=c\left(dA\right)$
Associative law for scalar multiplication

$1A=A$
Unit element for scalar multiplication

$c(A+B)=cA+cB$
Distributive law 1 for scalar multiplication

$(c+d)A=cA+dA$
Distributive law 2 for scalar multiplication
Properties of zero matrices
Then
the additive identity for all m×n matrices
the additive inverse of A
Transpose of a matrix
If a new matrix is formed by interchanging rows and columns the new matrix is called the transpose of the original matrix. For example, if:
Thus
If A = AT, then A is symmetric
Symmetric matrix
A square matrix A is symmetric if A = AT
Skewsymmetric matrix:
A square matrix A is skewsymmetric if AT = –A
TYPES OF MATRICES
NAME 
DESCRIPTION 
EXAMPLE 
Row matrix

A matrix with only 1 row

$\left(\begin{array}{cccc}3& 2& 1& 4\end{array}\right)$ 
Column matrix

A matrix with only I column

$\left(\begin{array}{c}2\\ 3\end{array}\right)$ 
Identity matrix

Diagonal matrix having each diagonal element equal to one (I)

$\left\begin{array}{cc}1& 0\\ 0& 1\end{array}\right$ 
Zero matrix

A matrix with all zero entries

$\left\begin{array}{cc}0& 0\\ 0& 0\end{array}\right$ 
Upper Triangular matrix

Square matrix having all the entries zero below the principal diagonal

$\left[\begin{array}{ccc}2& 5& 6\\ 0& 4& 6\\ 0& 0& 7\end{array}\right]$ 
Lower Triangular matrix

Square matrix having all the entries zero above the principal diagonal

$\left[\begin{array}{ccc}2& 0& 0\\ 5& 4& 0\\ 3& 6& 7\end{array}\right]$ 
Special matrices
 Square matrix
 Diagonal matrix
 Unit matrix (identity matrix)
 Null matrix
Square matrix
 A square matrix is of order m × m.

A square matrix is symmetric if For example:
$\left(\begin{array}{ccc}1& 2& 2\\ 2& 8& 9\\ 5& 9& 4\end{array}\right)$

A square matrix is skewsymmetric
if
For example
$\left(\begin{array}{ccc}0& 2& 5\\ 2& 0& 9\\ 5& 9& 0\end{array}\right)$
Diagonal Matrices.
A diagonal matrix is a square matrix that has values on the diagonal with all offdiagonal entities being zero
Identity Matrix (Unit Matrix)

An identity matrix is a diagonal matrix where the diagonal elements all equal one.
$I=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

It is named I and it comes in different sizes.
$I2=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$I3=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$
Null Matrix (Zero Matrix)
A square matrix where all elements equal zero
$\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
Properties of zero matrices
Matrix Multiplication
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cc}5& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{cc}5& 2\\ 3& 4\end{array}\right]$
$\left[\begin{array}{cc}5& 2\\ 3& 4\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=\left[\begin{array}{cc}5& 2\\ 3& 4\end{array}\right]$
So, the identity matrix multiplied by any matrix
lets the “any” matrix keep its identity!
Mathematically, IA = A and AI = A !!
The Multiplicative Identity
$AI=A$
Multiply
$\left[\begin{array}{cc}2& 5\\ 4& 0\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$
$A=\left[\begin{array}{cc}2& 5\\ 4& 0\end{array}\right]$
Give the multiplicative identity for matrix B.
$B=\left[\begin{array}{cccc}0& 7& 4& 9\\ 3& 7& 9& 2\\ 0& 1& 4& 7\\ 6& 0& 4& 1\end{array}\right]I=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
This identity matrix is I4.
Matrix Multiplication constraint
 (m × n) × (p × n) = cannot be done
 (1 × n) × (n × 1) = a scalar (1x1)

Matrix Multiplication Properties
 AB does not necessarily equal BA
 (BA may even be an impossible operation)
For example
A popular matrix
Matrix Multiplication Examples
Matrix Inverse
 Matrix Inverse: Needed to perform the “division” of 2 square matrices
 In scalar terms A/B is the same as A * 1/B
 When we want to divide matrix A by matrix B we simply multiply by A by the inverse of B
 An inverse matrix is defined as