Matrices are fundamental tools in electronics engineering, essential for analyzing and solving systems of linear equations that arise in circuit analysis, signal processing, and control systems. Mastery of matrices enables technicians to efficiently manage complex networks and transform signals using matrix operations. Kenyan electronics professionals frequently apply matrices to model multi-input multi-output (MIMO) systems and to perform state-space analysis for control design.
A matrix is a rectangular array of numbers arranged in rows and columns, used to represent data or coefficients in linear equations. Each element in a matrix is identified by its position, given by the row and column numbers. Matrices are denoted by uppercase letters such as \( A \), and elements by lowercase letters with two subscripts indicating their position, for example, \( a_{ij} \) is the element in the \( i^{th} \) row and \( j^{th} \) column of matrix \( A \).
$$ A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} $$
Example 1: Identify the element in the 2nd row, 3rd column of matrix
$$ A = \begin{bmatrix} 4 & 7 & 2 \\ 9 & 5 & 8 \\ 1 & 6 & 3 \end{bmatrix} $$
Given: Matrix \( A \)
Element at row 2, column 3 is \( a_{23} \).
$$ a_{23} = 8 $$
Answer: 8
Example 2: Write matrix \( B \) with 2 rows and 4 columns where \( b_{11} = 3 \), \( b_{12} = 0 \), \( b_{21} = -1 \), \( b_{24} = 5 \), and other elements zero.
Given: Positions and values
$$ B = \begin{bmatrix} 3 & 0 & 0 & 0 \\ -1 & 0 & 0 & 5 \end{bmatrix} $$
Answer:
Matrix \( B \) as above.
Example 3: Determine the size of matrix
$$ C = \begin{bmatrix} 2 & 5 & 1 \\ 4 & 6 & 3 \end{bmatrix} $$
Given: Matrix \( C \)
Number of rows \( m = 2 \), number of columns \( n = 3 \)
Answer: \( 2 \times 3 \) matrix
Matrix addition and subtraction are performed element-wise and require matrices of the same dimension. This operation is used in electronics when combining system matrices or adding signal vectors.
If
$$ A = [a_{ij}], B = [b_{ij}], \quad \text{both } m \times n, $$
then
$$ A + B = [a_{ij} + b_{ij}], \quad A - B = [a_{ij} - b_{ij}] $$
Example 1: Add matrices
$$ A = \begin{bmatrix} 1 & 3 \\ 4 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 0 \\ -1 & 3 \end{bmatrix} $$
Given: \( A, B \)
$$ A + B = \begin{bmatrix} 1 + 5 & 3 + 0 \\ 4 + (-1) & 2 + 3 \end{bmatrix} $$
$$ = \begin{bmatrix} 6 & 3 \\ 3 & 5 \end{bmatrix} $$
Answer:
$$ \begin{bmatrix} 6 & 3 \\ 3 & 5 \end{bmatrix} $$
Example 2: Subtract matrices
$$ C = \begin{bmatrix} 7 & 2 & 6 \\ 1 & 4 & 3 \end{bmatrix}, \quad D = \begin{bmatrix} 3 & 5 & 1 \\ 0 & 2 & 4 \end{bmatrix} $$
Given: \( C, D \)
$$ C - D = \begin{bmatrix} 7 - 3 & 2 - 5 & 6 - 1 \\ 1 - 0 & 4 - 2 & 3 - 4 \end{bmatrix} $$
$$ = \begin{bmatrix} 4 & -3 & 5 \\ 1 & 2 & -1 \end{bmatrix} $$
Answer:
$$ \begin{bmatrix} 4 & -3 & 5 \\ 1 & 2 & -1 \end{bmatrix} $$
Example 3: Verify if the following matrices can be added:
$$ E = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}, \quad F = \begin{bmatrix} 7 & 8 \\ 9 & 10 \end{bmatrix} $$
Given: Dimensions \( E: 2 \times 3 \), \( F: 2 \times 2 \)
Since dimensions are different, addition is not defined.
Answer: Matrices cannot be added
Scalar multiplication involves multiplying every element of a matrix by a constant scalar value. This operation is essential in electronics engineering for scaling signal vectors or system parameters.
If \( A = [a_{ij}] \) and \( k \) is a scalar, then
$$ kA = [k \times a_{ij}] $$
Example 1: Multiply matrix by scalar
$$ A = \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix}, \quad k = 4 $$
Given: \( A, k \)
$$ kA = 4 \times \begin{bmatrix} 2 & -1 \\ 0 & 3 \end{bmatrix} = \begin{bmatrix} 4 \times 2 & 4 \times (-1) \\ 4 \times 0 & 4 \times 3 \end{bmatrix} $$
$$ = \begin{bmatrix} 8 & -4 \\ 0 & 12 \end{bmatrix} $$
Answer:
$$ \begin{bmatrix} 8 & -4 \\ 0 & 12 \end{bmatrix} $$
Example 2: Scale signal matrix
Given a signal matrix
$$ S = \begin{bmatrix} 0.5 & 1.2 & -0.8 \\ 1.0 & -0.5 & 2.0 \end{bmatrix} $$
and scalar gain \( g = 3 \),
Calculate \( gS \).
$$ gS = 3 \times \begin{bmatrix} 0.5 & 1.2 & -0.8 \\ 1.0 & -0.5 & 2.0 \end{bmatrix} = \begin{bmatrix} 1.5 & 3.6 & -2.4 \\ 3.0 & -1.5 & 6.0 \end{bmatrix} $$
Answer:
$$ \begin{bmatrix} 1.5 & 3.6 & -2.4 \\ 3.0 & -1.5 & 6.0 \end{bmatrix} $$
Example 3: Multiply a matrix by zero
$$ M = \begin{bmatrix} 7 & 2 \\ -3 & 4 \end{bmatrix}, \quad k = 0 $$
$$ kM = 0 \times M = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} $$
Answer:
Zero matrix of the same size as \( M \).
Matrix multiplication combines two matrices to produce a third matrix, used extensively in electronics for transforming vectors, system state calculations, and network analysis. The number of columns of the first matrix must equal the number of rows of the second matrix.
If \( A \) is \( m \times n \) and \( B \) is \( n \times p \), then the product \( C = AB \) is \( m \times p \), where
$$ c_{ij} = \sum_{k=1}^n a_{ik} b_{kj} $$
Example 1: Multiply two matrices
$$ A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} $$
Given: \( A, B \)
Calculate \( AB \).
Calculate element \( c_{11} \):
$$ c_{11} = (1 \times 5) + (2 \times 7) = 5 + 14 = 19 $$
Calculate element \( c_{12} \):
$$ c_{12} = (1 \times 6) + (2 \times 8) = 6 + 16 = 22 $$
Calculate element \( c_{21} \):
$$ c_{21} = (3 \times 5) + (4 \times 7) = 15 + 28 = 43 $$
Calculate element \( c_{22} \):
$$ c_{22} = (3 \times 6) + (4 \times 8) = 18 + 32 = 50 $$
$$ AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} $$
Answer:
$$ \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix} $$
Example 2: Multiply non-square matrices
$$ C = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \end{bmatrix}, \quad D = \begin{bmatrix} 3 & 1 \\ 2 & 1 \\ 1 & 0 \end{bmatrix} $$
Given: \( C (2 \times 3), D (3 \times 2) \)
Calculate \( CD \) (size \( 2 \times 2 \)).
Calculate \( c_{11} \):
$$ c_{11} = (1 \times 3) + (0 \times 2) + (2 \times 1) = 3 + 0 + 2 = 5 $$
Calculate \( c_{12} \):
$$ c_{12} = (1 \times 1) + (0 \times 1) + (2 \times 0) = 1 + 0 + 0 = 1 $$
Calculate \( c_{21} \):
$$ c_{21} = (-1 \times 3) + (3 \times 2) + (1 \times 1) = -3 + 6 + 1 = 4 $$
Calculate \( c_{22} \):
$$ c_{22} = (-1 \times 1) + (3 \times 1) + (1 \times 0) = -1 + 3 + 0 = 2 $$
So,
$$ CD = \begin{bmatrix} 5 & 1 \\ 4 & 2 \end{bmatrix} $$
Answer:
$$ \begin{bmatrix} 5 & 1 \\ 4 & 2 \end{bmatrix} $$
Example 3: Verify if multiplication is possible for matrices
$$ E = \begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}, \quad F = \begin{bmatrix} 7 & 8 & 9 \end{bmatrix} $$
Given: \( E (3 \times 2), F (1 \times 3) \)
Number of columns of \( E = 2 \), number of rows of \( F = 1 \)
Since 2 ≠ 1, multiplication \( EF \) is not defined.
Answer: Multiplication not possible
Example 4: Applied electronics problem, state transformation matrix multiplication
An electronics engineer models system states with matrix
$$ X = \begin{bmatrix} 2 \\ -1 \end{bmatrix} $$
and system transformation matrix
$$ T = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} $$
Calculate the new state vector \( Y = TX \).
Calculate \( y_1 \):
$$ y_1 = (0 \times 2) + (1 \times -1) = 0 - 1 = -1 $$
Calculate \( y_2 \):
$$ y_2 = (-1 \times 2) + (0 \times -1) = -2 + 0 = -2 $$
$$ Y = \begin{bmatrix} -1 \\ -2 \end{bmatrix} $$
Answer:
$$ \begin{bmatrix} -1 \\ -2 \end{bmatrix} $$
Example 5: Challenge: Multiply a \( 3 \times 3 \) matrix by a \( 3 \times 1 \) vector
Given:
$$ A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \\ 4 & 2 & 0 \end{bmatrix}, \quad V = \begin{bmatrix} 3 \\ 5 \\ -2 \end{bmatrix} $$
Calculate \( AV \).
Calculate \( c_1 \):
$$ c_1 = (1 \times 3) + (0 \times 5) + (2 \times -2) = 3 + 0 - 4 = -1 $$
Calculate \( c_2 \):
$$ c_2 = (-1 \times 3) + (3 \times 5) + (1 \times -2) = -3 + 15 - 2 = 10 $$
Calculate \( c_3 \):
$$ c_3 = (4 \times 3) + (2 \times 5) + (0 \times -2) = 12 + 10 + 0 = 22 $$
Result:
$$ AV = \begin{bmatrix} -1 \\ 10 \\ 22 \end{bmatrix} $$
Answer:
$$ \begin{bmatrix} -1 \\ 10 \\ 22 \end{bmatrix} $$
Given matrices \( A = \begin{bmatrix} 2 & 4 \\ 1 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} \), calculate \( A + B \). (3 marks)
Multiply matrix \( C = \begin{bmatrix} 3 & -1 & 2 \\ 0 & 4 & 5 \end{bmatrix} \) by scalar \( k = 2 \). (3 marks)
Find the product of matrices \( D = \begin{bmatrix} 1 & 0 \\ 2 & 3 \end{bmatrix} \) and \( E = \begin{bmatrix} 4 & 1 \\ 5 & 2 \end{bmatrix} \). (4 marks)
Determine whether the product \( FG \) is defined if \( F \) is \( 3 \times 2 \) and \( G \) is \( 3 \times 4 \). (2 marks)
Calculate the matrix product for
$$ H = \begin{bmatrix} 2 & 1 & 3 \\ 0 & -1 & 4 \end{bmatrix}, \quad I = \begin{bmatrix} 1 & 0 \\ 2 & 1 \\ -1 & 3 \end{bmatrix} $$
(5 marks)
Create a free account to open more of this chapter.
Free: practical guides, quick cards, workplace scenarios and more.
Create a free accountThis chapter introduced the concept of a matrix as an array of numbers arranged in rows and columns. The order of a matrix was explained as the number of rows by the number of columns it contains. A special type of matrix called a square matrix was defined where the number of rows equals the number of columns. The conditions for compatibility in matrix addition and multiplication were discussed, emphasizing that matrices must have appropriate orders to perform these operations. The chapter covered multiplication of matrices as well as multiplication of a matrix by a scalar, detailing the procedures for each. The identity matrix was presented as a square matrix with ones on the main diagonal and zeros elsewhere, serving as the multiplicative identity in matrix algebra. The determinant of a 2 by 2 matrix was defined and calculated, which led to the concept of the inverse of a 2 by 2 matrix, including the condition for a matrix to be singular when the determinant is zero. Finally, the chapter demonstrated how to solve simultaneous equations in two unknowns using the matrix method and Cramer's Rule, applying the concepts of determinants and inverses to find solutions efficiently.
Given matrices \(A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}\), find \(A + B\). (2 marks)
Determine the order of the matrix \(C = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\). (1 mark)