
Applications of Matrices and Determinants


1.1 Introduction 

Zero Matrix 

Ex 1.1.1 (i) Find the Adjoint of 2 x 2 Matrix 

Ex 1.1.1 (ii) Find the Adjoint of 3 X 3 Matrix 

Ex 1.1.1 (iii) Adjoint of 3 X 3 Matrix 

Ex 1.1.2 (i) Find the Inverse of 2 x 2 Matrix 

Ex 1.1.2 (ii) Find the inverse of 3 X 3 Matrix 

Ex 1.1.2 (iii) Find the inverse of 3 X 3 Matrix 

Ex 1.1.3 If F(α) = , show that [F(α)]⁻¹ = F(−α). 

Ex 1.1.4 If A = , show that A² − 3A − 7I₂ = O₂. Hence find A⁻¹. 

Ex 1.1.5 If A = , prove that A⁻¹ = A^{T}. 

Ex 1.1.6 If A = , verify that A(adj A) = (adj A)A = ∣A∣ I₂. 

Ex 1.1.7 If A = and B = , verify that (AB)⁻¹ = B⁻¹ A⁻¹. 

Ex 1.1.8 If adj(A) = , find A. 

Ex 1.1.9 If adj(A) = , find A⁻¹. 

Ex 1.1.10 Find adj(adj(A)) if adj A = 

Ex 1.1.11 A = show that A^{T} A⁻¹ = 

Ex 1.1.12 Find the matrix A for which A = 

Ex 1.1.13 Given A = , B = find a matrix X such that AXB = C. 

Ex 1.1.14 If A = , show that A⁻¹ = 1/2(A² − 3I). 

Ex 1.1.15 Decrypt the received encoded message [2 −3][20 4] with the encryption matrix and the decryption matrix as its inverse, where the system of codes are described by the numbers 126 to the letters A − Z respectively, and the number 0 to a blank space. 

Unit Matrix 

1.2 Inverse of a NonSingular Square Matrix 

Ex 1.2.1 (i) Find the Rank of the Matrix Minor Method 

Ex 1.2.1 (ii) Find the Rank of the Matrix Minor Method 

Ex 1.2.1 (iii) Find the Rank of the Matrix Minor Method 

Ex 1.2.1 (iv) Find the Rank of the Matrix Minor Method 

Ex 1.2.1 (v) Find the Rank of the Matrix Minor Method 

Ex 1.2.2 (i) Find the Rank of the Matrix Row Reduction Method 

Ex 1.2.2 (ii) Find the Rank of the Matrix Row Reduction Method 

Ex 1.2.2 (iii) Find the Rank of the Matrix Row Reduction Method 

Ex 1.2.3 (i) Find the inverse of the Matrix Gauss  Jordan Method 

Ex 1.2.3 (ii) Find the inverse of the Matrix Gauss  Jordan Method 

Ex 1.2.3 (iii) Find the inverse of the Matrix Gauss  Jordan Method 

Row Echelon Form  Matrices & Determinants 

Transpose of a Matrix 

1.3 Elementary Transformations of a Matrix 

1.4 Applications of Matrices: Solving System of Linear Equations 

1.5 Applications of Matrices: Consistency of system of linear equations by rank method


2 Complex Numbers 

2.1 Introduction to Complex Numbers 

2.2 Complex Numbers 

2.3 Basic Algebraic Properties of Complex Numbers 

2.4 Conjugate of a Complex Number 

2.5 Modulus of a Complex Number 

2.6 Geometry and Locus of Complex Numbers 

2.7 Polar and Euler form of a Complex Number 

2.8 de Moivre's Theorem and its Applications 

3 Theory of Equations


3.1 Introduction 

3.2 Basics of Polynomial Equations 

3.3 Vieta’s Formulae and Formation of Polynomial Equations 

3.4 Nature of Roots and Nature of Coefficients of Polynomial Equations 

3.5 Applications to Geometrical Problems 

3.6 Roots of Higher Degree Polynomial Equations 

3.7 Polynomials with Additional Information 

3.8 Polynomial Equations with no additional information 

3.9 Descartes Rule 

4 Inverse Trigonometric Functions 

4.1 Introduction 

4.2 Some Fundamental Concepts 

4.3 Sine Function and Inverse Sine Function 

4.4 The Cosine Function and Inverse Cosine Function 

4.5 The Tangent Function and the Inverse Tangent Function 

4.6 The Cosecant Function and the Inverse Cosecant Function 

4.7 The Secant Function and Inverse Secant Function 

4.8 The Cotangent Function and the Inverse Cotangent Function 

4.9 Principal Value of Inverse Trigonometric Functions 

4.10 Properties of Inverse Trigonometric Functions 

5 Two Dimensional Analytical GeometryII 

5.1 Introduction 

5.2 Circle 

5.3 Conics 

5.4 Conic Sections 

5.5 Parametric form of Conics 

5.6 Tangents and Normals to Conics 

5.7 Real life Applications of Conics 

6 Applications of Vector Algebra 

6.1 Introduction 

6.2 Geometric Introduction to Vectors 

6.3 Scalar Product and Vector Product 

6.4 Scalar triple product 

6.5 Vector triple product 

6.6 Jacobi’s Identity and Lagrange’s Identity 

6.7 Different forms of Equation of a Straight line 

6.8 Different forms of Equation of a plane 

6.9 Image of a point in a plane 

6.10 Meeting point of a line and a plane 

7 Applications of Differential Calculus 

7.1 Introduction 

Ex 7.1.1 A particle moves along a straight line in such a way that after t seconds its distance from the origin is s = 2t² + 3t metres. (i) Find the average velocity between t = 3 and t = 6 seconds. (ii) Find the instantaneous velocities at t = 3 and t = 6 seconds. 

Ex 7.1.2 A camera is accidentally knocked off an edge of a cliff 400 ft high. The camera falls a distance of s=16 t² in t seconds. (i) How long does the camera fall before it hits the ground? (ii) What is the average velocity with which the camera falls during the last 2 seconds? (iii) What is the instantaneous velocity of the camera when it hits the ground? 

Ex 7.1.3 A particle moves along a line according to the law s(t) = 2t³ − 9t² + 12t − 4, where t ≥ 0. (i) At what times the particle changes direction? (ii) Find the total distance travelled by the particle in the first 4 seconds. (iii) Find the particle’s acceleration each time the velocity is zero. 

Ex 7.1.4 If the volume of a cube of side length x is v = x³. Find the rate of change of the volume with respect to x when x = 5 units. 

Ex 7.1.5 If the mass m(x) (in kilograms) of a thin rod of length x (in metres) is given by, m(x) = √3x then what is the rate of change of mass with respect to the length when it is x = 3 and x = 27 metres. 

Ex 7.1.6 A stone is dropped into a pond causing ripples in the form of concentric circles. The radius r of the outer ripple is increasing at a constant rate at 2 cm per second. When the radius is 5 cm find the rate of changing of the total area of the disturbed water? 

Ex 7.1.7 A beacon makes one revolution every 10 seconds. It is located on a ship which is anchored 5 km from a straight shore line. How fast is the beam moving along the shore line when it makes an angle of 45° with the shore? 

Ex 7.1.8 A conical water tank with vertex down of 12 metres height has a radius of 5 metres at the top. If water flows into the tank at a rate 10 cubic m/min, how fast is the depth of the water increases when the water is 8 metres deep? 

Ex 7.1.9 A ladder 17 metre long is leaning against the wall. The base of the ladder is pulled away from the wall at a rate of 5 m/s. When the base of the ladder is 8 metres from the wall, (i) how fast is the top of the ladder moving down the wall? (ii) at what rate, the area of the triangle formed by the ladder, wall, and the floor, is changing? 

Ex 7.1.10 A police jeep, approaching an orthogonal intersection from the northern direction, is chasing a speeding car that has turned and moving straight east. When the jeep is 0.6 km north of the intersection and the car is 0.8 km to the east. The police determine with a radar that the distance between them and the car is increasing at 20 km/hr. If the jeep is moving at 60 km/hr at the instant of measurement, what is the speed of the car? 

7.2 Meaning of Derivatives 

Ex 7.2.1 Find the slope of the tangent to the following curves at the respective given points. (i) y = x⁴ + 2x² − x at x = 1 (ii) x = a cos³ t, y = b sin³ t at t = π/2. 

Ex 7.2.2 Find the point on the curve y = x² − 5x + 4 at which the tangent is parallel to the line 3x + y = 7. 

Ex 7.2.3 Find the points on the curve y = x³ − 6x² + x + 3 where the normal is parallel to the line x + y = 1729. 

Ex 7.2.4 Find the points on the curve y² − 4xy = x² + 5 for which the tangent is horizontal. 

Find the tangent and normal to the following curves at the given points on the curve. Ex 7.2.5.1 y = x² − x⁴ at (1, 0) 

Ex 7.2.5.2 y = x⁴ + 2e^{x} at (0, 2) 

Ex 7.2.5.3 y = x sin x at (π/2, π/2) 

Ex 7.2.5.4 x = cos t, y = 2sin² t at t = π/3 

Ex 7.2.6 Find the equations of the tangents to the curve y = 1 + x³ for which the tangent is orthogonal with the line x + 12y = 12. 

Ex 7.2.7 Find the equations of the tangents to the curve y = x + 1/x  1 which are parallel to the line x + 2y = 6. 

Ex 7.2.8 Find the equation of tangent and normal to the curve given by x = 7 cos t and y = 2sin t, t ∈ ℝ at any point on the curve. 

Ex 7.2.10 Show that the two curves x² − y² = r² and xy = c² where c, r are constants, cut orthogonally. 

7.3 Mean Value Theorem 

7.4 Series Expansions 

7.5 Indeterminate Forms 

7.6 Applications of First Derivative 

7.7 Applications of Second Derivative 

7.8 Applications in Optimization 

7.9 Symmetry and Asymptotes 

7.10 Sketching of Curves 

8 Differentials and Partial Derivatives 

8.1 Introduction 

8.2 Linear Approximation and Differentials 

8.3 Functions of Several Variables 

8.4 Limit and Continuity of Functions of Two Variables 

8.5 Partial Derivatives 

8.6 Linear Approximation and Differential of a Function of Several Variables 

9 Applications of Integration 

9.1 Introduction 

9.2 Definite Integral as the Limit of a Sum 

9.3 Fundamental Theorems of Integral Calculus and their Applications 

9.4 Bernoulli’s Formula 

9.5 Improper Integrals 

9.6 Reduction Formulae 

9.7 Gamma Integral 

9.8 Evaluation of Bounded Plane Area by Integration 

9.9 Volume of a Solid obtained by Revolving Area about an Axis 

10 Ordinary Differential Equations 

10.1 Introduction 

10.2 Differential Equation, Order, and Degree 

10.3 Classification of Differential Equations 

10.4 Formation of Differential Equations 

10.5 Solution of Ordinary Differential Equations 



10.6 Solution of First Order and First Degree Differential Equations 

10.7 First Order Linear Differential Equations 

10.8 Applications of First Order Ordinary Differential Equations 

11 Probability Distributions 

11.1 Introduction 

11.2 Random Variable 

11.3 Types of Random Variable 

11.4 Continuous Distributions 

11.5 Mathematical Expectation 

11.6 Theoretical Distributions: Some Special Discrete Distributions 

12 Discrete Mathematics 

12.1 Introduction 

12.2 Binary Operations 

12.3 Mathematical Logic 