Refrigeration and air conditioning professionals often manage systems involving multiple interconnected components whose behaviors can be modeled and analyzed using matrices. Mastery of matrix operations is essential for solving systems of linear equations, analyzing networks, and optimizing system performance in refrigeration and air conditioning design and troubleshooting. This chapter introduces matrix operations fundamental to engineering mathematics, focusing on addition, subtraction, and multiplication, with worked examples contextualized to refrigeration and air conditioning scenarios in Kenya.

1.1 Matrix operation

1.1.1 Addition

Matrix addition involves combining two matrices by adding their corresponding elements. This operation is applicable only when the matrices have the same dimensions, a common scenario when summing heat load matrices or adding system parameter arrays in refrigeration system analysis.

The formula for matrix addition is:

$$ C = A + B $$

where matrices \(A\) and \(B\) are of order \(m \times n\), and matrix \(C\) is the resulting matrix of the same order with elements

$$ c_{ij} = a_{ij} + b_{ij} $$

Worked Examples

Example 1: A refrigeration technician records heat gains from two different sources in a cold room as matrices \(A\) and \(B\):

Given:

$$ A = \begin{bmatrix} 120 & 150 \\ 80 & 100 \end{bmatrix} \text{ W}, \quad B = \begin{bmatrix} 30 & 50 \\ 20 & 40 \end{bmatrix} \text{ W} $$

Find the total heat gains matrix \(C\).

$$ C = A + B $$

$$ C = \begin{bmatrix} 120 + 30 & 150 + 50 \\ 80 + 20 & 100 + 40 \end{bmatrix} $$

$$ C = \begin{bmatrix} 150 & 200 \\ 100 & 140 \end{bmatrix} \text{ W} $$

Answer:

\[ \boxed{ \begin{bmatrix} 150 & 200 \\ 100 & 140 \end{bmatrix} \text{ W} } \]

Example 2: Two air conditioning units have airflow rate matrices (in m³/min) as follows:

$$ A = \begin{bmatrix} 5 & 7 & 6 \\ 4 & 6 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 2 & 3 \end{bmatrix} $$

Calculate the combined airflow rates matrix \(C\).

$$ C = A + B $$

$$ C = \begin{bmatrix} 5+2 & 7+3 & 6+4 \\ 4+1 & 6+2 & 5+3 \end{bmatrix} $$

$$ C = \begin{bmatrix} 7 & 10 & 10 \\ 5 & 8 & 8 \end{bmatrix} \text{ m}^3/\text{min} $$

Answer:

\[ \boxed{ \begin{bmatrix} 7 & 10 & 10 \\ 5 & 8 & 8 \end{bmatrix} \text{ m}^3/\text{min} } \]

Example 3: A refrigeration system's pressure drops across components are recorded as:

$$ A = \begin{bmatrix} 1.2 & 0.8 & 1.0 \\ 0.9 & 1.1 & 1.3 \end{bmatrix} \text{ bar}, \quad B = \begin{bmatrix} 0.3 & 0.2 & 0.5 \\ 0.4 & 0.3 & 0.2 \end{bmatrix} \text{ bar} $$

Find the total pressure drop matrix.

$$ C = A + B $$

$$ C = \begin{bmatrix} 1.2+0.3 & 0.8+0.2 & 1.0+0.5 \\ 0.9+0.4 & 1.1+0.3 & 1.3+0.2 \end{bmatrix} $$

$$ C = \begin{bmatrix} 1.5 & 1.0 & 1.5 \\ 1.3 & 1.4 & 1.5 \end{bmatrix} \text{ bar} $$

Answer:

\[ \boxed{ \begin{bmatrix} 1.5 & 1.0 & 1.5 \\ 1.3 & 1.4 & 1.5 \end{bmatrix} \text{ bar} } \]

1.1.2 Subtraction

Matrix subtraction is the process of deducting one matrix from another by subtracting corresponding elements. It is used in refrigeration engineering to calculate differences such as net heat load after accounting for heat losses or to determine changes in system parameters.

The formula for matrix subtraction is:

$$ D = A - B $$

where matrices \(A\) and \(B\) are of the same order \(m \times n\), and the elements of matrix \(D\) are

$$ d_{ij} = a_{ij} - b_{ij} $$

Worked Examples

Example 1: A cold storage facility measures heat gains and heat losses as matrices \(A\) and \(B\):

Given:

$$ A = \begin{bmatrix} 250 & 300 \\ 180 & 220 \end{bmatrix} \text{ W}, \quad B = \begin{bmatrix} 50 & 70 \\ 40 & 60 \end{bmatrix} \text{ W} $$

Calculate the net heat gain matrix \(D\).

$$ D = A - B $$

$$ D = \begin{bmatrix} 250 - 50 & 300 - 70 \\ 180 - 40 & 220 - 60 \end{bmatrix} $$

$$ D = \begin{bmatrix} 200 & 230 \\ 140 & 160 \end{bmatrix} \text{ W} $$

Answer:

\[ \boxed{ \begin{bmatrix} 200 & 230 \\ 140 & 160 \end{bmatrix} \text{ W} } \]

Example 2: Airflow rates (in m³/min) before and after filter cleaning are given by:

$$ A = \begin{bmatrix} 12 & 15 & 13 \\ 10 & 14 & 11 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 3 & 2 \\ 1 & 2 & 1 \end{bmatrix} $$

Find the increase in airflow rate matrix after cleaning.

$$ D = A - B $$

$$ D = \begin{bmatrix} 12 - 2 & 15 - 3 & 13 - 2 \\ 10 - 1 & 14 - 2 & 11 - 1 \end{bmatrix} $$

$$ D = \begin{bmatrix} 10 & 12 & 11 \\ 9 & 12 & 10 \end{bmatrix} \text{ m}^3/\text{min} $$

Answer:

\[ \boxed{ \begin{bmatrix} 10 & 12 & 11 \\ 9 & 12 & 10 \end{bmatrix} \text{ m}^3/\text{min} } \]

Example 3: Pressure readings (in bar) at two points in a refrigeration pipeline before and after maintenance are:

$$ A = \begin{bmatrix} 2.5 & 3.0 \\ 2.8 & 3.2 \end{bmatrix}, \quad B = \begin{bmatrix} 0.5 & 0.6 \\ 0.4 & 0.7 \end{bmatrix} $$

Calculate the pressure difference matrix.

$$ D = A - B $$

$$ D = \begin{bmatrix} 2.5 - 0.5 & 3.0 - 0.6 \\ 2.8 - 0.4 & 3.2 - 0.7 \end{bmatrix} $$

$$ D = \begin{bmatrix} 2.0 & 2.4 \\ 2.4 & 2.5 \end{bmatrix} \text{ bar} $$

Answer:

\[ \boxed{ \begin{bmatrix} 2.0 & 2.4 \\ 2.4 & 2.5 \end{bmatrix} \text{ bar} } \]

1.1.3 Multiplication

Matrix multiplication combines two matrices to produce a third matrix, representing composition of linear transformations or system interactions. In refrigeration and air conditioning, it is used to calculate system responses, such as heat transfer through multiple stages or airflow distribution through duct networks.

Matrix multiplication is defined when matrix \(A\) is of order \(m \times n\) and matrix \(B\) is of order \(n \times p\). The resulting matrix \(E\) is of order \(m \times p\) with elements:

$$ e_{ij} = \sum_{k=1}^n a_{ik} b_{kj} $$

Worked Examples

Example 1: A refrigeration system has a matrix \(A\) representing heat transfer coefficients (W/m²·K) and matrix \(B\) representing temperature differences (K):

Given:

$$ A = \begin{bmatrix} 10 & 5 \\ 3 & 8 \end{bmatrix}, \quad B = \begin{bmatrix} 15 \\ 20 \end{bmatrix} $$

Calculate the heat transfer matrix \(E\).

$$ E = A \times B $$

Calculate each element:

$$ e_{11} = (10 \times 15) + (5 \times 20) = 150 + 100 = 250 $$

$$ e_{21} = (3 \times 15) + (8 \times 20) = 45 + 160 = 205 $$

So,

$$ E = \begin{bmatrix} 250 \\ 205 \end{bmatrix} \text{ W/m}^2 $$

Answer:

\[ \boxed{ \begin{bmatrix} 250 \\ 205 \end{bmatrix} \text{ W/m}^2 } \]

Example 2: Airflow distribution matrix \(A\) (m³/min per zone) and control matrix \(B\) (control factors) are:

$$ A = \begin{bmatrix} 4 & 7 & 6 \\ 3 & 5 & 8 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 1 & 1 \end{bmatrix} $$

Find the resulting airflow matrix \(E\).

Calculate each element:

$$ e_{11} = (4 \times 1) + (7 \times 0) + (6 \times 1) = 4 + 0 + 6 = 10 $$

$$ e_{12} = (4 \times 0) + (7 \times 1) + (6 \times 1) = 0 + 7 + 6 = 13 $$

$$ e_{21} = (3 \times 1) + (5 \times 0) + (8 \times 1) = 3 + 0 + 8 = 11 $$

$$ e_{22} = (3 \times 0) + (5 \times 1) + (8 \times 1) = 0 + 5 + 8 = 13 $$

So,

$$ E = \begin{bmatrix} 10 & 13 \\ 11 & 13 \end{bmatrix} \text{ m}^3/\text{min} $$

Answer:

\[ \boxed{ \begin{bmatrix} 10 & 13 \\ 11 & 13 \end{bmatrix} \text{ m}^3/\text{min} } \]

Example 3: A refrigeration system's state matrix \(A\) and transformation matrix \(B\) are:

$$ A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 0 \\ 1 & 3 \\ 0 & 1 \end{bmatrix} $$

Calculate the product matrix \(E\).

Calculate elements:

$$ e_{11} = (1 \times 2) + (2 \times 1) + (3 \times 0) = 2 + 2 + 0 = 4 $$

$$ e_{12} = (1 \times 0) + (2 \times 3) + (3 \times 1) = 0 + 6 + 3 = 9 $$

$$ e_{21} = (0 \times 2) + (1 \times 1) + (4 \times 0) = 0 + 1 + 0 = 1 $$

$$ e_{22} = (0 \times 0) + (1 \times 3) + (4 \times 1) = 0 + 3 + 4 = 7 $$

$$ E = \begin{bmatrix} 4 & 9 \\ 1 & 7 \end{bmatrix} $$

Answer:

\[ \boxed{ \begin{bmatrix} 4 & 9 \\ 1 & 7 \end{bmatrix} } \]

Example 4: A system matrix \(A\) of order \(3 \times 2\) and matrix \(B\) of order \(2 \times 3\) are:

$$ A = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}, \quad B = \begin{bmatrix} 7 & 8 & 9 \\ 10 & 11 & 12 \end{bmatrix} $$

Calculate the product matrix \(E\) of order \(3 \times 3\).

Calculate each element:

$$ e_{11} = (1 \times 7) + (4 \times 10) = 7 + 40 = 47 $$

$$ e_{12} = (1 \times 8) + (4 \times 11) = 8 + 44 = 52 $$

$$ e_{13} = (1 \times 9) + (4 \times 12) = 9 + 48 = 57 $$

$$ e_{21} = (2 \times 7) + (5 \times 10) = 14 + 50 = 64 $$

$$ e_{22} = (2 \times 8) + (5 \times 11) = 16 + 55 = 71 $$

$$ e_{23} = (2 \times 9) + (5 \times 12) = 18 + 60 = 78 $$

$$ e_{31} = (3 \times 7) + (6 \times 10) = 21 + 60 = 81 $$

$$ e_{32} = (3 \times 8) + (6 \times 11) = 24 + 66 = 90 $$

$$ e_{33} = (3 \times 9) + (6 \times 12) = 27 + 72 = 99 $$

$$ E = \begin{bmatrix} 47 & 52 & 57 \\ 64 & 71 & 78 \\ 81 & 90 & 99 \end{bmatrix} $$

Answer:

\[ \boxed{ \begin{bmatrix} 47 & 52 & 57 \\ 64 & 71 & 78 \\ 81 & 90 & 99 \end{bmatrix} } \]

Chapter Summary

This chapter introduced fundamental matrix operations including addition, subtraction, and multiplication, establishing the basis for manipulating matrices in engineering problems. It then focused on calculating the determinant and inverse of a 2x2 matrix, essential tools for solving linear systems. Using these concepts, methods to find solutions of linear simultaneous equations with two unknowns were demonstrated. The study advanced to determinants and inverses of 3x3 matrices, providing techniques to handle more complex systems of equations. The chapter also covered solving linear simultaneous equations involving three unknowns using matrix methods. Finally, it explored practical applications of matrices in engineering contexts, highlighting their importance in modelling and problem-solving. Through this progression, students gained comprehensive skills to apply matrices effectively in various engineering scenarios.