Matrices and Surds, Determinants, Inverse and Simplification for SS 3

Class: Senior Secondary School 3 (SS3 / SSS3)
Term: 1st Term
Week: 4
Age: 17 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Algebra and Trigonometry
Previous Lesson: Matrices, Definition, Types and Operations
Topic: MATRICES – II / SURDS
Subject Matter: Transpose of a matrix, determinant of 2×2 and 3×3 matrices, solution of simultaneous equations using determinant method, inverse of 2×2 matrix, conjugate of binomial surds, simplification of surds including difference of two squares in denominator, application to solving triangles involving special angles 30°, 45° and 60°, evaluation of expressions involving surds

Specific Objectives

By the end of the lesson, students should be able to:

Cognitive Domain:

• Define and find the transpose of a given matrix.
• Calculate the determinant of 2×2 and 3×3 matrices.
• Solve simultaneous linear equations using the determinant method (Cramer’s Rule).
• Determine the inverse of a 2×2 matrix.
• Identify the conjugate of binomial surds and use it for simplification.
• Simplify surds, including rationalizing denominators.
• Evaluate expressions involving surds.
• Apply surd values to solve problems involving special angles in trigonometry.

Affective Domain:

• Appreciate the importance of matrices and surds in solving mathematical problems.
• Develop a systematic approach to solving problems involving matrices and surds.
• Show interest in using mathematical techniques for real-world applications.

Psychomotor Domain:

• Accurately compute the transpose, determinant, and inverse of matrices.
• Correctly apply Cramer’s Rule to solve simultaneous equations.
• Skillfully simplify surd expressions and rationalize denominators.
• Precisely solve trigonometric problems involving special angles using surd values.

Social Domain:

• Collaborate with peers to solve complex mathematical problems.
• Communicate mathematical ideas clearly when discussing solutions.

Reference Materials

The following resources were used in planning this lesson:

• FCT-ERC SS 3 Scheme of Work, September 2014 curriculum.
• Relevant State Unified Scheme of Work.
• A suitable General Mathematics textbook for SS 3.

Instructional Materials

The teacher will teach this lesson with the aid of:

• Matrix charts showing examples of transpose, determinant, and inverse.
• Determinant charts illustrating Cramer’s Rule.
• Surds worksheets with various simplification and application problems.
• Computer instructional materials (if available) for visual demonstrations.
• Whiteboard and markers/chalk.

Rationale for the Lesson

This lesson helps students understand important mathematical tools like matrices and surds. These concepts are useful for solving systems of equations and simplifying complex expressions, which are applied in various fields like engineering, computer science, and physics. Understanding these topics also builds a strong foundation for advanced mathematics.

Prerequisite/Previous Knowledge

Students should have prior knowledge of basic matrix operations (addition, subtraction, multiplication), basic surd concepts (definition, addition, subtraction of like surds), and fundamental trigonometric ratios.

Lesson Content/Board Summary

MATRICES – II

Transpose of a Matrix

The transpose of a matrix is a new matrix formed by interchanging the rows and columns of the original matrix. If a matrix is denoted by A, its transpose is denoted by Aᵀ or A’.

Example 1:

If A = [[2, 3],
[1, 4]]

Then Aᵀ = [[2, 1],
[3, 4]]

Example 2:

If B = [[1, 2, 3],
[4, 5, 6]]

Then Bᵀ = [[1, 4],
[2, 5],
[3, 6]]

Determinant of a 2×2 Matrix

The determinant of a 2×2 matrix is a scalar value calculated from its elements. For a matrix A = [[a, b], [c, d]], the determinant is given by the formula:

det(A) = |A| = ad – bc

Example: Find the determinant of A = [[3, 5], [1, 2]]

Solution:

Step 1: Identify the elements a, b, c, d.
a = 3, b = 5, c = 1, d = 2

Step 2: Apply the formula ad – bc.
det(A) = (3 * 2) – (5 * 1)

Step 3: Calculate the result.
det(A) = 6 – 5 = 1

Determinant of a 3×3 Matrix

The determinant of a 3×3 matrix can be calculated by expanding along any row or column using minors and cofactors. For a matrix A = [[a, b, c], [d, e, f], [g, h, i]], expanding along the first row gives:

det(A) = a(ei – fh) – b(di – fg) + c(dh – eg)

Example: Find the determinant of A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

Solution:

Step 1: Use the expansion formula along the first row.

det(A) = 1 * det([[5, 6], [8, 9]]) – 2 * det([[4, 6], [7, 9]]) + 3 * det([[4, 5], [7, 8]])

Step 2: Calculate the 2×2 determinants.

det([[5, 6], [8, 9]]) = (5 * 9) – (6 * 8) = 45 – 48 = -3

det([[4, 6], [7, 9]]) = (4 * 9) – (6 * 7) = 36 – 42 = -6

det([[4, 5], [7, 8]]) = (4 * 8) – (5 * 7) = 32 – 35 = -3

Step 3: Substitute back into the 3×3 determinant formula.

det(A) = 1(-3) – 2(-6) + 3(-3)

Step 4: Calculate the final result.

det(A) = -3 + 12 – 9 = 0

Solution of Simultaneous Equations using Determinant Method (Cramer’s Rule)

Cramer’s Rule uses determinants to solve systems of linear equations. For a system of two linear equations:

ax + by = c
dx + ey = f

The values of x and y are given by:

x = Dx / D

y = Dy / D

Where:

1. D is the determinant of the coefficient matrix: D = det([[a, b], [d, e]])
2. Dx is the determinant formed by replacing the x-coefficient column in D with the constant terms: Dx = det([[c, b], [f, e]])
3. Dy is the determinant formed by replacing the y-coefficient column in D with the constant terms: Dy = det([[a, c], [d, f]])

Example: Solve the following simultaneous equations using Cramer’s Rule:

2x + 3y = 7
x – y = 1

Solution:

Step 1: Find D (determinant of coefficient matrix).

D = det([[2, 3], [1, -1]]) = (2 * -1) – (3 * 1) = -2 – 3 = -5

Step 2: Find Dx (replace x-coefficients with constants).

Dx = det([[7, 3], [1, -1]]) = (7 * -1) – (3 * 1) = -7 – 3 = -10

Step 3: Find Dy (replace y-coefficients with constants).

Dy = det([[2, 7], [1, 1]]) = (2 * 1) – (7 * 1) = 2 – 7 = -5

Step 4: Calculate x and y.

x = Dx / D = -10 / -5 = 2

y = Dy / D = -5 / -5 = 1

The solution is x = 2, y = 1.

Inverse of a 2×2 Matrix

The inverse of a square matrix A, denoted by A⁻¹, is a matrix such that when multiplied by A, it yields the identity matrix (I). An inverse exists only if the determinant of the matrix is not zero.

For a 2×2 matrix A = [[a, b], [c, d]], the inverse is given by the formula:

A⁻¹ = (1 / det(A)) * [[d, -b], [-c, a]]

Example: Find the inverse of A = [[3, 1], [4, 2]]

Solution:

Step 1: Find the determinant of A.

det(A) = (3 * 2) – (1 * 4) = 6 – 4 = 2

Step 2: If det(A) ≠ 0, proceed. (Here, 2 ≠ 0, so an inverse exists).

Step 3: Form the adjugate matrix by swapping a and d, and negating b and c.

Adjugate(A) = [[2, -1], [-4, 3]]

Step 4: Multiply the adjugate matrix by (1 / det(A)).

A⁻¹ = (1/2) * [[2, -1], [-4, 3]]

Step 5: Perform the scalar multiplication.

A⁻¹ = [[1, -1/2], [-2, 3/2]]

SURDS

Conjugate of Binomial Surds

The conjugate of a binomial surd is formed by changing the sign between the two terms. When a binomial surd is multiplied by its conjugate, the result is a rational number (no surd). This uses the “difference of two squares” property: (x + y)(x – y) = x² – y².

The following are examples of binomial surds and their conjugates:

1. The conjugate of (a + √b) is (a – √b).
2. The conjugate of (√a + √b) is (√a – √b).
3. The conjugate of (a – √b) is (a + √b).
4. The conjugate of (√a – √b) is (√a + √b).

Example 1: Find the product of (2 + √3) and its conjugate.

Solution:

The conjugate of (2 + √3) is (2 – √3).

(2 + √3)(2 – √3) = 2² – (√3)² = 4 – 3 = 1

Example 2: Find the product of (√5 – √2) and its conjugate.

Solution:

The conjugate of (√5 – √2) is (√5 + √2).

(√5 – √2)(√5 + √2) = (√5)² – (√2)² = 5 – 2 = 3

Simplification of Surds (including rationalizing the denominator)

Simplifying surds means expressing them in their simplest form, where the number under the square root sign has no perfect square factors other than 1. Rationalizing the denominator involves removing any surds from the denominator of a fraction.

Example 1 (Basic Simplification): Simplify √72

Solution:

Step 1: Find the largest perfect square factor of 72.
72 = 36 * 2

Step 2: Rewrite the surd using the factors.
√72 = √(36 * 2) = √36 * √2

Step 3: Simplify the perfect square.
√72 = 6√2

Example 2 (Rationalizing a Monomial Denominator): Simplify 5/√3

Solution:

Step 1: Multiply the numerator and denominator by the surd in the denominator.
5/√3 = (5/√3) * (√3/√3)

Step 2: Perform the multiplication.
= 5√3 / (√3 * √3) = 5√3 / 3

Example 3 (Rationalizing a Binomial Denominator): Simplify 1 / (2 + √3)

Solution:

Step 1: Multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of (2 + √3) is (2 – √3).

1 / (2 + √3) = (1 / (2 + √3)) * ((2 – √3) / (2 – √3))

Step 2: Multiply the numerators and denominators.

= (1 * (2 – √3)) / ((2 + √3)(2 – √3))

Step 3: Simplify the denominator using the difference of two squares.
= (2 – √3) / (2² – (√3)²) = (2 – √3) / (4 – 3)

Step 4: Calculate the final result.
= (2 – √3) / 1 = 2 – √3

Application to Solving Triangles involving Special Angles (30°, 45°, 60°)

Surds are used to represent the exact values of trigonometric ratios for special angles (30°, 45°, 60°). These exact values are important for solving problems involving right-angled triangles without using calculators.

The following are the exact trigonometric values for special angles:

1. For 30°: sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3 = √3/3
2. For 45°: sin 45° = 1/√2 = √2/2, cos 45° = 1/√2 = √2/2, tan 45° = 1
3. For 60°: sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

Example 1: In a right-angled triangle, if the hypotenuse is 8 cm and one angle is 30°, find the length of the side opposite the 30° angle.

Solution:

Step 1: Identify the knowns and unknowns.
Hypotenuse = 8 cm, Angle = 30°, Unknown = Opposite side (let’s call it x)

Step 2: Choose the appropriate trigonometric ratio.
Since we have opposite and hypotenuse, use sine: sin θ = Opposite / Hypotenuse

Step 3: Substitute the values.
sin 30° = x / 8

Step 4: Use the exact value of sin 30°.
1/2 = x / 8

Step 5: Solve for x.
2x = 8
x = 4 cm

Example 2: A ladder leans against a wall, making an angle of 60° with the ground. If the base of the ladder is 5 meters from the wall, how high up the wall does the ladder reach?

Solution:

Step 1: Identify the knowns and unknowns.
Angle = 60°, Adjacent side (distance from wall) = 5 m, Unknown = Opposite side (height, let’s call it h)

Step 2: Choose the appropriate trigonometric ratio.
Since we have opposite and adjacent, use tangent: tan θ = Opposite / Adjacent

Step 3: Substitute the values.
tan 60° = h / 5

Step 4: Use the exact value of tan 60°.
√3 = h / 5

Step 5: Solve for h.
h = 5√3 meters

Evaluation of Expressions involving Surds

This involves performing arithmetic operations (addition, subtraction, multiplication, division) and simplifying expressions that contain surds.

Example 1: Evaluate (√5 + √2)(√5 – √2)

Solution:

Step 1: Recognize this as a difference of two squares (x + y)(x – y) = x² – y².
(√5 + √2)(√5 – √2) = (√5)² – (√2)²

Step 2: Simplify the squares.
= 5 – 2

Step 3: Calculate the final result.
= 3

Example 2 : Evaluate (3√2 + √3)²

Solution:

Step 1: Apply the identity (a + b)² = a² + 2ab + b²

= (3√2)² + 2(3√2)(√3) + (√3)²

Step 2: Simplify each term.

(3√2)² = 9 × 2 = 18
2(3√2)(√3) = 6√6
(√3)² = 3

Step 3: Add the results.

= 18 + 6√6 + 3

Final Answer: 21 + 6√6

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Quick Revision Table

Area	Key Point
Transpose of a Matrix	The transpose of a matrix is a new matrix formed by interchanging the rows and columns of the original matrix. If a matrix is denoted by A, its…
Determinant of a 2×2 Matrix	The determinant of a 2×2 matrix is a scalar value calculated from its elements. For a matrix A = [[a, b], [c, d]], the determinant is given by the…
Determinant of a 3×3 Matrix	The determinant of a 3×3 matrix can be calculated by expanding along any row or column using minors and cofactors. For a matrix A = [[a, b, c],…
Solution of Simultaneous Equations using Determinant Method (Cramer's Rule)	Cramer's Rule uses determinants to solve systems of linear equations. For a system of two linear equations: ax + by = c dx + ey = f The values of…
Inverse of a 2×2 Matrix	The inverse of a square matrix A, denoted by A⁻¹, is a matrix such that when multiplied by A, it yields the identity matrix (I). An inverse exists…
Conjugate of Binomial Surds	The conjugate of a binomial surd is formed by changing the sign between the two terms. When a binomial surd is multiplied by its conjugate, the…

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Guided Practice, Question and Answer, Visual Aids.

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Instructional Procedures

Step 1: Introduction and Recall

Time: 4 minutes

Teaching Skill: Set Induction

Teacher’s Activity: The teacher greets students, checks attendance, and revises previous matrix and surd concepts. Introduces today’s topic.

Students’ Activity: Students respond to questions and prepare for the lesson.

Learning Point: Prior matrix knowledge.

Step 2: Matrices (Transpose, Determinant, Inverse, Cramer’s Rule)

Time: 5 minutes

Teaching Skill: Explanation / Demonstration

Teacher’s Activity: The teacher explains transpose, determinants (2×2 and 3×3), inverse of 2×2 matrices, and solving simultaneous equations using Cramer’s Rule with worked examples.

Students’ Activity: Students follow explanations, copy examples, and attempt guided problems.

Learning Point: Matrix operations and methods.

Step 3: Surds (Conjugates and Simplification)

Time: 5 minutes

Teaching Skill: Demonstration

Teacher’s Activity: The teacher explains conjugates, rationalization of denominators, and simplification of surds using worked examples.

Students’ Activity: Students practice simplifying surds and rationalizing denominators.

Learning Point: Surd simplification methods.

Step 4: Application to Special Angles

Time: 5 minutes

Teaching Skill: Guided Practice

Teacher’s Activity: The teacher demonstrates how surd values are applied in solving triangles involving 30°, 45°, and 60°.

Students’ Activity: Students solve sample problems using exact trigonometric values.

Learning Point: Matrix practical application.

Step 5: Guided Practice

Time: 5 minutes

Teaching Skill: Guided Practice/Problem Solving

Teacher’s Activity: The teacher gives two additional problems drawn from matrix and guides students to choose the correct method, show each stage of working and check their answers.

Students’ Activity: Students solve the guided problems, compare their working with the demonstrated method and correct any errors.

Learning Point: Matrix guided practice.

Step 6: Application and Reinforcement

Time: 5 minutes

Teaching Skill: Individual Practice/Reinforcement

Teacher’s Activity: The teacher presents a different matrix problem and asks students to identify the relevant rule or formula before solving it.

Students’ Activity: Students identify the appropriate method, solve the problem independently and check the reasonableness of the answer.

Learning Point: Matrix practical application.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Find the determinant of a given 2×2 matrix.
2. Solve a pair of equations using Cramer’s Rule.
3. Rationalize a surd denominator.
4. Evaluate a simple surd expression.

Students’ Activity: Students answer orally and in writing, showing working or giving examples where the lesson requires it.

Learning Point: Matrix skills assessed.

Step 8: Note-Taking

Time: 8 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides students to copy the essential Board Summary on matrix into their notebooks, drawing attention to the key rules, examples, diagrams or distinctions already taught.

Students’ Activity: Students copy the essential notes carefully into their notebooks and correct any errors in their earlier work.

Learning Point: Matrix notes recorded.

Step 9: Conclusion

Time: 3 minutes

Teaching Skill: Reinforcement/Summarization

Teacher’s Activity: The teacher summarizes key points.

Students’ Activity: Students listen.

Learning Point: Matrix concepts consolidated.

Continuous Assessment/Further Study

Type: Practice Exercise

Instruction: Complete the following tasks in your notebook. Show all mathematical working clearly.

1. Find the transpose of \(A=\begin{pmatrix}2&3\\-1&4\end{pmatrix}\).
2. Find the determinant of \(\begin{pmatrix}5&2\\3&4\end{pmatrix}\).
3. Find the inverse of \(\begin{pmatrix}2&1\\1&1\end{pmatrix}\), if it exists.
4. Rationalise the denominator of \(\frac{3}{\sqrt{5}+2}\).
5. Simplify \(\sqrt{75}-2\sqrt{3}+\sqrt{12}\).

Lesson Keywords

• Transpose – Interchanging rows and columns of a matrix.
• Determinant – A scalar value derived from a square matrix.
• Inverse Matrix – A matrix which produces the identity matrix when multiplied by the original.
• Conjugate – A binomial surd formed by changing the sign between terms.
• Surd – An irrational number expressed with a root sign.
• Rationalization – Removing surds from denominators.

Differentiation

Fast learners attempt mixed matrix–surd problems independently.
Average learners solve guided examples with minimal assistance.
Slow learners receive step-by-step support on determinants, conjugates, and basic surd simplification.

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