Application of Algebra

Collaborative Learning, Experiential Learning, Talk for Learning, problem-based Learning
Activity 1: Revise Types of Matrices Collaborative Learning, Experiential Learning, Whole class discussion, Talk for Learning Approaches and Problem-based Learning. Learners work in pairs; one of the pairs identifies a matrix they know whiles the other partially writes an example (learners' reverse roles).
Example: Revise special types of Matrices with examples (i.e. Square matrix, Diagonal matrix, Identity Matrix, Upper Triangular matrix, Lower Triangular Matrix, Symmetric Matrix, Skew-Symmetric Matrix, Zero Matrix, Row Vector, Column Vector)
Activity 2: Revise Matrices Algebra Collaborative Learning, Experiential Learning, Whole class discussion, Talk for Learning Approaches and Problem-based Learning.
Learners work in pairs: one pair identifies and creates a problem involving operation on a matrix and tasks the other pair to solve, justify, prove or investigate the context (learners' reverse roles).
Example
- Learners revise Equality of matrices, Scalar multiple of a matrix, Addition of two matrices, Multiplication of two matrices and Properties of Matrices (commutative, Associative and Distributive law).
Activity 3: Determinant, transpose of a matrix, singular and non-singular matrix, Minor Cofactors of a square matrix.
Collaborative Learning, Experiential Learning, Whole class discussion, Talk for Learning Approaches and Problem-based Learning.
- Work with a partner: one of the pair identifies a create a 2x2 and 3x3 matrix whiles the other pair writes the transpose of the matrix and verify the transpose properties. Learners change roles.
- Learners establish that the transpose of matrix A, written A t is the matrix obtained by writing the rows of A in order as columns.
Example 2 −1 2 4 j then 𝐴 G = i If 𝐴 = i j 4 3 −1 3
- Work with a partner: one of the pair identifies and creates a 2x2 matrix while the other pair calculate the determinant of the matrix. Learners change roles.
𝑎 𝑏
- Establish that; If 𝐴 = i j then the determinant of matrix A, denoted by det(A) or |𝐴| is given 𝑐 𝑑 by |𝐴| = 𝑎 − 𝑏.
Note: If the determinant of a matrix is zero, then that matrix is called a singular matrix.
Example 3 −5 Find the determinant of 𝐴 = i j 2 −4
Solution: |𝐴| = 3 × (−4) − (−5) × 2 = −12 + 10 = −2
Example 4 8 Find the determinant of 𝐵 = i j 3 6
Solution: |𝐵| = 4 × (6) − 8 × (3) = 24 − 24 = 0 Matrix B is singular because its determinant is zero.
- If A is a square matrix, verify the following properties of the Determinant: i) The determinant of a square matrix A and its transpose are equal. |𝐴| = ⌊𝐴 G ⌋. ii) If A has a row (column) of zeros, then |𝐴| = 0. iii) If A has two identical rows (or columns), then |𝐴| = 0. iv) If A is a triangular matrix, then |𝐴| is a product of the diagonal elements. v) If A is a square matrix of order 𝑛 and 𝑘 is a scalar, then |𝑘| = 𝑘 % ⌊𝐴⌋.
- Work with a partner: One pair identifies and creates a 2x2 matrix while the other pair investigates whether the matrix is singular or non-singular. Learners change roles and summarise their findings.
Example: The table shows a comparison of singular and non-singular matrix
Non-singular Singular A is invertible not invertible Columns independent dependent Rows independent dependent det(A) ≠ 0 = 0 Ax = 0 one solution x = 0 infinitely many solutions or Ax = b one solution no solution
Teaching and Learning Resources:
- Worksheets
- Scientific Calculator
- Technological tools, apps, etc.
Assessment (2.1.1.AS.1). The document marks these depth-of-knowledge levels for this indicator: Level 3 Strategic reasoning.