1. Matrices and matrix operations - Mathematics for Business

1.1Introduction¶

Example

A certain economy is comprised of the following three sectors: agriculture, manufacturing and energy. Each industry consumes a percentage of outputs from the other industries in order to produce its goods, and those percentages are given in the following table:

\begin{array}{| c | c | c | c |} \hline \hspace{0.2 cm} & \text{Agriculture} &\text{Manufacturing} &\text{Energy} \\ \hline \text{Agriculture} &0.2 &0.3 &0.1 \\ \hline \text{Manufacturing} &0.4 &0.1 &0.2 \\ \hline \text{Energy} &0.3 &0.2 &0.2 \\ \hline \end{array}

The final demands for these industries are given in the following table:

\begin{array}{| c | c |} \hline \text{Agriculture} &100\\ \hline \text{Manufacturing} &150 \\ \hline \text{Energy} &80\\ \hline \end{array}

How much should each industry produce in order to meet the demands?

Example

A company tracks customer behaviour where people move between three kinds of customers: loyal, occasional and lost. The probabilities that a certain customer will change its category in a time period of half a year are given in the following table:

\begin{array}{|c|c|c|c|} \hline \text{From/To } &\text{Loyal} &\text{Occasional} &\text{Lost} \\ \hline \text{Loyal} &0.7 &0.1 &0.1 \\ \hline \text{Occasional} &0.2 &0.6 &0.2 \\ \hline \text{Lost} &0.1 &0.3 &0.7 \\ \hline \end{array}

If we start with a thousand customers, which are categorized in the following way:

\begin{array}{| c | c |} \hline \text{Loyal} &600 \\ \hline \text{Occasional} &300 \\ \hline \text{Lost} &100 \\ \hline \end{array}

determine the distribution of these customers a year from now.

There are two ways in which we can specifiy a matrix:

you can write it down explicitly
you can specify its order and state the formula for computing its entries

1.2Types of matrices¶

1.3Matrix operations¶

Tip

When defining any operation which involves matrices, you need to answer the following questions:

What are the inputs of the operation?
What is the results of the operation?
How do we compute the result?

Transpose of a matrix¶

Matrix addition¶

Multiplication by a scalar¶

Scalar product¶

Matrix multiplication¶

Animation showcasing how we compute a product of two conformable matrices.

Mathematics for Business

Part 1. Linear algebra

2. Determinant of a matrix