Matrix Algebra – An Introduction & Basic Operations
Why Matrices Algebra
The discovery of Matrices and Determinant returns back to 18th & 19th centuries it was initially used for solving problems of
- Transformation of geometric objects.
- Systems of linear equations
After that with the spread of the computers and the rapid development of computer technology the matrix’s algebra became an important basic for many of computer since related issues , we can state here some of them
- Graph Theory.
- Theory of Games.
- Computer Graphics and Computed Tomography
- Cryptography.
So we can conclude that every Software Engineer must know this very important branch of algebra.
Basic Operations :
Let’s consider the following case to know start dealing with Matrix , let’s consider 2 companies have expenses that need to be recorded monthly like salaries, Running Costs,Cost of Goods,…. let’s limit this to salaries and Running Costs and Taxes , Now how would one represent the data collected , many ways exists to do this but the Matrix has the advantage of combining the data so it’s easy to manipulate them , so we writes the data as follow
and if we are not confusing about what this values is for we can write it as follow
this is what called Matrix , size is the matrix is defined by two factors number of Rows and number of Columns as in our example the above matrix has 2 rows and 3 columns i.e 2×3 matrix as general we represent the matrix size on the form nxm where n is the number of rows and the m is the number of columns
for our example if we need to represent the 1st 3 mouthes data this could be a follow
now if we need to calculate the first quarter companies expenses what we would do ?
for example for salaries for company 1 the result could be 1000 + 1200 + 800 ,here we can come to one of Matrices operation which is the Matrices’s addition which is done by adding the entries one by one as follow
now what can we do to double the matrix ? clearly we can add it to itself as follow
which implies
So we can conclude the following rule
for any constant x we can write
Which is the Matrecies’s operation Called Matrix Constant Multiplication or Mmultiplication of a Matrix By a Number , keeping in mind we write the constant to the left of the matrices in the case of multiplication.
now we come to the Subtraction Operation it’s a combination of the above 2 rules i.e
M-N=M+(-1)N
so we multiply N with -1 then add it to M.
Examples :
Consider the following Matrices
Lets calculate the following values
- 2M1 – M2
- M3+M1-3M2
- M3+M2
