Matrices are essential tools in various professional fields, including healthcare, business, agriculture, and ICT, as they provide a structured way to organize and manipulate data. Understanding matrices enables efficient handling of systems of equations, data transformations, and optimization problems encountered in county hospitals, banks, farms, and government offices. This chapter introduces the fundamentals of matrices and their operations, forming the foundation for more advanced applications in engineering mathematics.

9.1 Introduction to matrices

Matrices are rectangular arrays of numbers arranged in rows and columns. They are used to represent data, solve systems of linear equations, and perform transformations in multiple disciplines. This section covers the definition, types, notation, and basic operations on matrices.

9.1.1 Definition of a matrix

A matrix is a rectangular array of elements (usually numbers) arranged in rows and columns. The size of a matrix is described by the number of rows \(m\) and columns \(n\), denoted as an \(m \times n\) matrix. Matrices are used to represent data or coefficients in linear systems.

$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$

Worked Examples

Example 1: A county hospital records the number of patients in three departments over four days as a matrix:

Given: Number of patients matrix \(P\) is a \(3 \times 4\) matrix.

$$ P = \begin{bmatrix} 5 & 7 & 6 & 8 \\ 3 & 4 & 5 & 6 \\ 2 & 3 & 4 & 3 \end{bmatrix} $$

Identify the number of rows and columns.

$$ \text{Rows} = 3, \quad \text{Columns} = 4 $$

Answer: Matrix \(P\) is a \(3 \times 4\) matrix with 3 rows and 4 columns.

Example 2: A SACCO tracks monthly loan applications in two branches over three months:

Given:

$$ L = \begin{bmatrix} 12 & 15 & 10 \\ 8 & 9 & 7 \end{bmatrix} $$

Determine matrix size.

$$ \text{Rows} = 2, \quad \text{Columns} = 3 $$

Answer: Matrix \(L\) is a \(2 \times 3\) matrix.

Example 3: A retail business lists sales in three product categories for five days:

Given matrix \(S\):

$$ S = \begin{bmatrix} 20 & 25 & 30 & 28 & 26 \\ 15 & 18 & 22 & 20 & 19 \\ 10 & 12 & 15 & 14 & 13 \end{bmatrix} $$

Find matrix size.

$$ \text{Rows} = 3, \quad \text{Columns} = 5 $$

Answer: Matrix \(S\) is a \(3 \times 5\) matrix.

9.1.2 Different types of matrices: row, column, square, rectangular, diagonal, identity, zero

Matrices are classified based on their dimensions and elements. Understanding these types helps in selecting appropriate operations.

• Row matrix: Has one row and multiple columns (\(1 \times n\)).
• Column matrix: Has one column and multiple rows (\(m \times 1\)).
• Square matrix: Number of rows equals number of columns (\(n \times n\)).
• Rectangular matrix: Number of rows differs from number of columns.
• Diagonal matrix: Square matrix with non-zero elements only on the main diagonal.
• Identity matrix: Diagonal matrix with all diagonal elements equal to 1.
• Zero matrix: All elements are zero.

Worked Examples

Example 1: Identify the type of matrix:

$$ R = \begin{bmatrix} 4 & 7 & 5 \end{bmatrix} $$

Since there is 1 row and 3 columns, \(R\) is a row matrix.

Answer: \(R\) is a row matrix.

Example 2: Identify matrix type:

$$ C = \begin{bmatrix} 3 \\ 6 \\ 2 \end{bmatrix} $$

Since there are 3 rows and 1 column, \(C\) is a column matrix.

Answer: \(C\) is a column matrix.

Example 3: Given matrix

$$ Q = \begin{bmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{bmatrix} $$

This is a square matrix with non-zero elements only on the diagonal, so \(Q\) is a diagonal matrix.

Answer: \(Q\) is a diagonal matrix.

Example 4: Given matrix

$$ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

This is a diagonal matrix with diagonal elements all 1, so it is an identity matrix.

Answer: \(I\) is an identity matrix.

Example 5: Given matrix

$$ Z = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} $$

All elements are zero, so \(Z\) is a zero matrix.

Answer: \(Z\) is a zero matrix.

9.1.3 Notation and elements of a matrix

Matrix elements are denoted by \(a_{ij}\), where \(i\) is the row number and \(j\) is the column number. The size of the matrix is \(m \times n\) representing \(m\) rows and \(n\) columns.

$$ A = [a_{ij}] \quad \text{where} \quad i = 1,2,\ldots,m; \quad j = 1,2,\ldots,n $$

The element \(a_{23}\) is the element in the 2nd row and 3rd column.

Worked Examples

Example 1: For matrix

$$ M = \begin{bmatrix} 2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix} $$

Find element \(a_{12}\).

$$ a_{12} = 4 $$

Answer: The element in row 1, column 2 is 4.

Example 2: For matrix

$$ N = \begin{bmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{bmatrix} $$

Find element \(a_{31}\).

$$ a_{31} = 11 $$

Answer: The element in row 3, column 1 is 11.

Example 3: For matrix

$$ P = \begin{bmatrix} 5 & 0 & 1 & 2 \\ 3 & 6 & 4 & 7 \end{bmatrix} $$

Find element \(a_{24}\).

$$ a_{24} = 7 $$

Answer: The element in row 2, column 4 is 7.

9.1.4 Basic operations: addition, subtraction, scalar multiplication

Matrices can be added, subtracted, or multiplied by scalars under specific rules.

• Addition: Only matrices of the same size can be added by adding corresponding elements.

$$ C = A + B, \quad c_{ij} = a_{ij} + b_{ij} $$

• Subtraction: Same rule as addition, subtract corresponding elements.

$$ D = A - B, \quad d_{ij} = a_{ij} - b_{ij} $$

• Scalar multiplication: Multiply every element by a scalar \(k\).

$$ E = kA, \quad e_{ij} = k \times a_{ij} $$

Worked Examples

Example 1: Addition of two matrices representing stock in a retail store:

$$ A = \begin{bmatrix} 10 & 25 \\ 15 & 30 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 10 \\ 8 & 12 \end{bmatrix} $$

$$ C = A + B $$

$$ c_{11} = 10 + 5 = 15, \quad c_{12} = 25 + 10 = 35 $$

$$ c_{21} = 15 + 8 = 23, \quad c_{22} = 30 + 12 = 42 $$

$$ C = \begin{bmatrix} 15 & 35 \\ 23 & 42 \end{bmatrix} $$

Answer: Matrix \(C\) is

$$ \begin{bmatrix} 15 & 35 \\ 23 & 42 \end{bmatrix} $$

Example 2: Subtracting two matrices representing daily water usage (in cubic metres) at a farm:

$$ A = \begin{bmatrix} 100 & 120 \\ 90 & 110 \end{bmatrix}, \quad B = \begin{bmatrix} 20 & 30 \\ 15 & 25 \end{bmatrix} $$

$$ D = A - B $$

$$ d_{11} = 100 - 20 = 80, \quad d_{12} = 120 - 30 = 90 $$

$$ d_{21} = 90 - 15 = 75, \quad d_{22} = 110 - 25 = 85 $$

$$ D = \begin{bmatrix} 80 & 90 \\ 75 & 85 \end{bmatrix} $$

Answer: Matrix \(D\) is

$$ \begin{bmatrix} 80 & 90 \\ 75 & 85 \end{bmatrix} $$

Example 3: Scalar multiplication of a matrix representing monthly electricity consumption (in kWh) at a county office:

$$ A = \begin{bmatrix} 200 & 250 \\ 180 & 210 \end{bmatrix} $$

Multiply by scalar \(k = 1.05\) to account for 5% increase.

$$ E = 1.05 \times A $$

$$ e_{11} = 1.05 \times 200 = 210, \quad e_{12} = 1.05 \times 250 = 262.5 $$

$$ e_{21} = 1.05 \times 180 = 189, \quad e_{22} = 1.05 \times 210 = 220.5 $$

$$ E = \begin{bmatrix} 210 & 262.5 \\ 189 & 220.5 \end{bmatrix} $$

Answer: Matrix \(E\) is

$$ \begin{bmatrix} 210 & 262.5 \\ 189 & 220.5 \end{bmatrix} $$

9.1.5 Special Matrices

Special matrices have unique properties useful in mathematical operations.

• Symmetric matrix: Equal to its transpose, \(A = A^T\).
• Skew-symmetric matrix: Transpose equals negative, \(A^T = -A\).
• Upper triangular matrix: All elements below the main diagonal are zero.
• Lower triangular matrix: All elements above the main diagonal are zero.
• Orthogonal matrix: Inverse equals transpose, \(A^{-1} = A^T\).

Worked Examples

Example 1: Check if matrix is symmetric:

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

Transpose \(A^T\):

$$ A^T = \begin{bmatrix} 4 & 2 & 3 \\ 2 & 5 & 1 \\ 3 & 1 & 6 \end{bmatrix} $$

Since \(A = A^T\), \(A\) is symmetric.

Answer: Matrix \(A\) is symmetric.

Example 2: Check if matrix is skew-symmetric:

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

Transpose \(B^T\):

$$ B^T = \begin{bmatrix} 0 & -3 & 2 \\ 3 & 0 & -5 \\ -2 & 5 & 0 \end{bmatrix} $$

Check if \(B^T = -B\):

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

Since \(B^T = -B\), \(B\) is skew-symmetric.

Answer: Matrix \(B\) is skew-symmetric.

Example 3: Identify if matrix is upper triangular:

$$ C = \begin{bmatrix} 1 & 4 & 7 \\ 0 & 5 & 8 \\ 0 & 0 & 9 \end{bmatrix} $$

All elements below the main diagonal are zero, so \(C\) is upper triangular.

Answer: Matrix \(C\) is upper triangular.

Example 4: Identify if matrix is lower triangular:

$$ D = \begin{bmatrix} 3 & 0 & 0 \\ 2 & 6 & 0 \\ 4 & 5 & 7 \end{bmatrix} $$

All elements above the main diagonal are zero, so \(D\) is lower triangular.

Answer: Matrix \(D\) is lower triangular.

Example 5: Verify if matrix \(E\) is orthogonal:

$$ E = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} $$

Calculate transpose:

$$ E^T = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $$

Calculate inverse \(E^{-1}\):

Since \(E\) is \(2 \times 2\), inverse is

$$ E^{-1} = \frac{1}{\det E} \begin{bmatrix} e_{22} & -e_{12} \\ -e_{21} & e_{11} \end{bmatrix} $$

Calculate determinant:

$$ \det E = (0)(0) - (1)(-1) = 0 + 1 = 1 $$

Calculate inverse:

$$ E^{-1} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} $$

Since \(E^{-1} = E^T\), \(E\) is orthogonal.

Answer: Matrix \(E\) is orthogonal.

Chapter Summary

This chapter introduced matrices by defining them and describing their various types including row, column, square, rectangular, diagonal, identity, and zero matrices. The notation and elements of a matrix were explained, followed by basic operations such as addition, subtraction, and scalar multiplication. Special matrices like identity, diagonal, symmetric, and skew-symmetric matrices were also covered. Matrix multiplication was explored with emphasis on its rules and key properties including associativity, commutativity, and transpose characteristics. The chapter then focused on determinants, detailing how to calculate them for 2x2 and 3x3 matrices, and methods to find inverses using the adjoint and Gauss-Jordan elimination techniques, alongside conditions for an inverse to exist. Systems of linear equations were represented using matrices, and methods for solving these systems were presented, including Cramer's rule and the inverse matrix method. Finally, the inverse determinant method was applied as another approach to solve linear systems effectively.