Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Vectors and Matrices → Matrices ↓


Inverse of a Matrix


The concept of the inverse of a matrix is important in mathematics, particularly in the field of linear algebra. Let's dig deeper to understand the inverse of a matrix, including definitions, properties, how it is calculated, and some examples.

Introduction to matrices

Before knowing the inverse of a matrix, it is important to understand what a matrix is. A matrix is essentially a rectangular array of numbers arranged in rows and columns. The individual items in a matrix are called elements or entries.

For example, a 2x2 matrix looks like this:

    | ab | | cd |
    | ab | | cd |

Here, a, b, c and d are the elements of the matrix. The size of this matrix is 2x2, which means it has 2 rows and 2 columns.

Definition of the inverse of a matrix

The inverse of a matrix is the matrix that when multiplied by the original matrix gives the identity matrix. The identity matrix is a special type of matrix that has 1's on the diagonal and 0's elsewhere. For a 2x2 matrix, the identity matrix is:

    | 1 0 | | 0 1 |
    | 1 0 | | 0 1 |

If A is a square matrix, then its inverse is denoted by A-1, and it satisfies the following condition:

    A * A-1 = I

where I is the identity matrix of the same size as A

Conditions for a matrix to be inverse

Not all matrices have an inverse. For a matrix to have an inverse, two main conditions must be met:

  1. The matrix must be a square matrix (same number of rows and columns).
  2. The determinant of the matrix must not be zero.

Determinant of a matrix

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix:

    | ab | | cd |
    | ab | | cd |

The determinant (denoted by det(A) or |A|) is given as:

    det(A) = ad - bc

Finding the inverse of a 2x2 matrix

If A is a 2x2 matrix:

    | ab | | cd |
    | ab | | cd |

The inverse of a matrix A, if it exists, is given by:

    A-1 = (1/det(A)) * | d -b | | -ca |
    A-1 = (1/det(A)) * | d -b | | -ca |

provided that det(A) ≠ 0.

Example calculation

Consider the matrix:

    A = | 2 3 | | 1 4 |
    A = | 2 3 | | 1 4 |

First, calculate the determinant:

    det(A) = 2*4 - 3*1 = 8 - 3 = 5

Since the determinant is not zero, the inverse exists. The inverse is A-1:

    A-1 = (1/5) * | 4 -3 | | -1 2 | A-1 = | 0.8 -0.6 | | -0.2 0.4 |
    A-1 = (1/5) * | 4 -3 | | -1 2 | A-1 = | 0.8 -0.6 | | -0.2 0.4 |

Multiplicative properties

When a matrix is multiplied by its inverse, the result is the identity matrix, as shown in the previous definition. Using the example above:

    A * A-1 = | 2 3 | * | 0.8 -0.6 | | 1 4 | | -0.2 0.4 | = | (2*0.8 + 3*-0.2) (2*-0.6 + 3*0.4) | | (1*0.8 + 4*-0.2) (1*-0.6 + 4*0.4) | = | 1.0 0.0 | | 0.0 1.0 |
    A * A-1 = | 2 3 | * | 0.8 -0.6 | | 1 4 | | -0.2 0.4 | = | (2*0.8 + 3*-0.2) (2*-0.6 + 3*0.4) | | (1*0.8 + 4*-0.2) (1*-0.6 + 4*0.4) | = | 1.0 0.0 | | 0.0 1.0 |

The result is the identity matrix.

Visual explanations

Let us try to understand the concept of inverse with a graphical representation. Consider a line segment in a two-dimensional plane:

    
    
    
    In Verse
    Original

Imagine turning this blue line into a red line using matrix operations. When the inverse matrix is then applied, it will bring the red line back to its original blue state.

Conclusion

The inverse of a matrix is a powerful tool in linear algebra, and it has many applications in solving systems of linear equations, computer graphics, and more. Understanding how to calculate and apply the inverse matrix is essential for exploring deeper concepts in mathematics.

While this lesson focused on 2x2 matrices, the same principles apply to larger matrices as well; however, calculating the inverse becomes more complicated as the size of the matrix increases. Nevertheless, the basic principle is that the inverse of a matrix satisfies the following condition:

    A * A-1 = I

This concludes our comprehensive journey through the concept of the inverse of a matrix using simple mathematical expressions and visual examples to provide clarity to beginners stepping into the field of Linear Algebra.


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Inverse of a Matrix

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