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 ↓


Determinants


Determinants are essential in linear algebra, which is associated with matrices. The determinant is a special number that is calculated from a square matrix. Its calculation provides valuable information about the matrix and the system of equations it represents.

Understanding determinants

Let us first clarify what a square matrix is. A square matrix is one in which the number of rows is equal to the number of columns. The determinant helps us in various ways, such as determining whether a matrix is invertible or not or understanding the properties of linear transformations.

Determinant of a 2x2 matrix

For a 2x2 matrix, calculating the determinant is quite simple. Consider the following matrix:

A = | ab | | cd |

The determinant of the matrix A is given by the formula:

det(A) = ad - bc

For clarity, let's look at an example:

A = | 4 3 | | 2 1 | det(A) = (4 * 1) - (3 * 2) = 4 - 6 = -2
(4,3) (2,1) det = -2

Determinant of a 3x3 matrix

For a 3x3 matrix, the determinant is a bit more complicated. Consider a 3x3 matrix:

B = | abc | | def | | ghi |

The determinant of the matrix B is calculated as follows:

det(B) = a(ei − fh) − b(di − fg) + c(dh − eg)

Here's an example:

B = | 1 2 3 | | 4 5 6 | | 7 8 9 | det(B) = 1(5*9 - 6*8) - 2(4*9 - 6*7) + 3(4*8 - 5*7) = 1(45 - 48) - 2(36 - 42) + 3(32 - 35) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0
(1,2,3) (4,5,6) (7,8,9) det = 0

Properties of determinants

Understanding the properties of determinants can make complex calculations much simpler. Here are some key properties:

1. A row or column of zeros

If all the elements in a row or column of a matrix are zero, then the determinant of that matrix will be zero.

C = | 1 2 3 | | 0 0 0 | | 7 8 9 | det(C) = 0

2. Swapping rows or columns

If any two rows or any two columns of a matrix are interchanged then the determinant changes its sign.

3. Multiplying a row or column by a scalar

If you multiply a row or column of a matrix by a scalar, the determinant of the matrix gets multiplied by that scalar.

4. The sum of two determinants

The determinant of a matrix is linear with respect to its rows and columns. If you add the coefficient of one row or column to another row or column, the determinant of the matrix does not change.

Applications of determinants

Determinants have various applications in mathematics and related fields. Knowing how to handle determinants is important for the following areas:

1. Solving systems of linear equations

Determinants can be used to solve systems of linear equations using Cramer's rule, which applies to systems of equations where the number of equations equals the number of unknowns.

2. Finding the inverse of matrices

If the determinant of a matrix is not zero, then the matrix has an inverse. The inverse of a matrix is important in solving linear algebra equations, especially those that are expressed in matrix form.

3. Calculation of area and volume

When the coordinates of the vertices are known, determinants can be used to calculate the areas of triangles or parallelograms and the volumes of parallelograms or prisms.

Area = 0.5 * base * height

Conclusion

The concept of determinants is a fundamental topic in the study of matrices. It helps determine the properties of matrices and the solution sets of the systems they represent. Learning to apply these rules and properties allows solving problems in linear algebra and other related mathematical fields more effectively.

Practicing problems involving determinants can further enhance understanding and provide skills required in higher level mathematics and various applications in physics, engineering, and economics.


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Determinants

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