Answer to Question #195969 in Linear Algebra for EUGINE HAWEZA

Question

Write notes on how to add, subtract and multiply matrices, and show how they may apply to solving a three system of equation , site examples on the applications of matrices to solving real world business problems

Answer 1

Addition of Matrices

The addition of matrices is done on an entry by entry mode. This implies that before you can add up two matrices together, you have to be sure that they are of the same order (i.e they have the same number of rows and columns).

Consider,

$A = \begin{bmatrix} 1&3&4\\ 2&-2&1 \end{bmatrix}$

and

$B = \begin{bmatrix} 5&0&-10\\ 3&-2&1 \end{bmatrix}$

We see that both A and B are 2by3 matrices (i.e they both have 2 rows and 3 columns).

Now let's see what A+B will look like.

$A+B =\begin{bmatrix} 1+5&3+0&4-10\\2+3&-2-2&1+1 \end{bmatrix} = \begin{bmatrix} 6&3&-6\\5&-4&2 \end{bmatrix}$

Substraction of Matrices

The operation of substraction of matrices is similar to that of addition of matrices. It is also done on an entry by entry mode.

Using the same matrices A and B used under addition of matrices, consider

$A-B =\begin{bmatrix} 1-5&3-0&4+10\\2-3&-2+2&1-1 \end{bmatrix} = \begin{bmatrix} -4&3&14\\-1&0&0\end{bmatrix}$

Multiplication of matrices

The idea of multiplication of matrices has to do with compatibility. What do I mean by this, two matrices are said to be compatible if the number of column of the first matrix is equal to the row of the second matrix. It is also important to note that normal algebraic addition will be employed in getting the multiplication of two matrices. Consider the example below

Let $C = \begin{bmatrix} -2&1&3\\ 2&0&5 \end{bmatrix}$

and

$D = \begin{bmatrix} -3\\5\\4\end{bmatrix}$

Matrix C has 2 rows and 3 columns and a matrix D has 3 rows and 1 column. So we have that matrix C and D are compatible. And it is expected that matrix CD will have 2 rows and 1 column. Let's see this for ourselves.

$CD = \begin{bmatrix} -2(-3)+1(5)+3(4)\\2(-3)+0(5)+5(4) \end{bmatrix} = \begin{bmatrix} 6+5+12\\-6+0+20\end{bmatrix}$

$= \begin{bmatrix} 23\\14 \end{bmatrix}$

We see that matrix DC does not exist because matrix D has 3 rows and 1 column while matrix C has 2 rows and 3 columns. Hence they are not compatible and their product is not defined. In general, we have that $CD \neq DC$ for all matrices C and D.

System of 3 equations

Consider the example

$3x +2y-z = 1 \\ 2x-4y+5z=9 \\ x+2y-8z =6$

This can be expressed in matrices form as

$\begin{bmatrix} 3&2&-1\\2&-4&5\\1&2&-8 \end{bmatrix} \begin{bmatrix}x\\y\\z \end{bmatrix} = \begin{bmatrix} 1\\9\\6\end{bmatrix}$

The idea of multiplication of matrices is used in getting this matrix expression.

Application of matrices in solving real world business problems

1) Matrices are the brain box behind security systems. Every business owner wants their transactions to be secured and matrices are used to create encrypted documents that keep security systems safe.

2) Matrices are used in the concept of Markov chains and this helps business make informed decision based on certain properties that are well laid out.

3) Matrices are also used in assignment problem. Assignment problem are problems that have to do with assigning particular task to particular facilities with the broad view of maximizing profit and minimizing cost.

4) They are also used in transportation problems

5)They are used in creating models that help out in the decision making of firms and organizations.