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 ↓


Cramer's Rule


Cramer's rule is a mathematical theorem used to solve systems of linear equations that contain equivalent equations for the unknowns, provided that the system has a unique solution. It gives an explicit formula for the solution in terms of determinants. Cramer's rule is named after the Swiss mathematician Gabriel Cramer.

Basics of determinants

Before understanding Cramer's rule, let us understand determinants, which are an important component.

For a 2x2 matrix:

|ab| |cd|

The determinant is calculated as follows:

det = ad - bc

For a 3x3 matrix:

|abc| |def| |ghi|

The determinant is calculated as follows:

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

Understanding Cramer's rule

Consider a system of n linear equations with n unknowns. In matrix form, the system can be written as:

AX = B

Where:

  • A is an nxn matrix of coefficients
  • X is a column matrix containing variables [x1, x2, ..., xn]
  • B is a column matrix of constants [b1, b2, ..., bn]

If the determinant of the matrix A (denoted as det(A)) is not zero, then the system has a unique solution. Cramer's rule provides the solution as follows:

xi = det(Ai) / det(A) for i = 1, 2, ..., n

where Ai is the matrix A, but its i-th column is replaced by the matrix B.

Step-by-step example

Example 1: Solving a 2x2 system

Consider the following system of equations:

2x + 3y = 8 x - y = 1

In matrix form:

A = | 2 3 | | 1 -1 | X = | x | | y | B = | 8 | | 1 |

To find det(A):

det(A) = (2)(-1) - (3)(1) = -2 - 3 = -5

Now, calculate the matrices A1 and A2:

A1 = | 8 3 | | 1 -1 |
det(A1) = (8)(-1) - (3)(1) = -8 - 3 = -11
A2 = | 2 8 | | 1 1 |
det(A2) = (2)(1) - (8)(1) = 2 - 8 = -6

Use of Cramer's Rule:

x = det(A1)/det(A) = -11 / -5 = 2.2 y = det(A2)/det(A) = -6 / -5 = 1.2

Example 2: Solving a 3x3 system

Consider these equations:

x + 2y + 3z = 9 2x + 3y + z = 8 3x + y + 2z = 7

Matrix form:

A = | 1 2 3 | | 2 3 1 | | 3 1 2 | X = | x | | y | | z | B = | 9 | | 8 | | 7 |

Calculate the determinant of A:

det(A) = 1(3*2 - 1*2) - 2(2*2 - 1*3) + 3(2*1 - 3*3) = 1(6 - 2) - 2(4 - 3) + 3(2 - 9) = 1*4 - 2*1 + 3*(-7) = 4 - 2 - 21 = -19

Let's calculate the matrices A1, A2 and A3:

A1 = | 9 2 3 | | 8 3 1 | | 7 1 2 |
det(A1) = 9(3*2 - 1*2) - 2(8*2 - 1*7) + 3(8*1 - 3*7) = 9(6 - 2) - 2(16 - 7) + 3(8 - 21) = 9*4 - 2*9 + 3*(-13) = 36 - 18 - 39 = -21
A2 = | 1 9 3 | | 2 8 1 | | 3 7 2 |
det(A2) = 1(8*2 - 1*7) - 9(2*2 - 1*3) + 3(2*1 - 3*7) = 1(16 - 7) - 9(4 - 3) + 3(2 - 21) = 1*9 - 9*1 + 3*(-19) = 9 - 9 - 57 = -57
A3 = | 1 2 9 | | 2 3 8 | | 3 1 7 |
det(A3) = 1(3*7 - 8*1) - 2(2*7 - 8*3) + 9(2*1 - 3*3) = 1(21 - 8) - 2(14 - 24) + 9(2 - 9) = 13 + 20 + (-63) = -30

Solve using Cramer's rule:

x = det(A1)/det(A) = -21 / -19 = 1.105 y = det(A2)/det(A) = -57 / -19 = 3 z = det(A3)/det(A) = -30 / -19 = 1.579

Important aspects of Cramer's rule

Cramer's rule is beautiful and clear, suitable for educational purposes and for small systems of equations. However, it is not efficient for large systems due to the computational intensity of the determinant calculations as the matrix size grows.

Applying to large matrices

Though theoretically applicable to any nxn system, the computational burden is apparently impractical for n > 3. For n = 4, or larger, standard methods such as Gaussian elimination or LU decomposition are preferred, mainly in practical applications.

Determinant cancellation

If det(A) = 0, then Cramer's rule cannot be applied because division by zero is undefined. Such cases often suggest infinite solutions or no solutions at all. Alternative methods will explore these aspects by considering row minimization techniques or matrix rank.

Visualization of examples

Assuming det(A) on nxn is non-zero, imagine solving via row operations so that the conjugation X is equivalent to Euclidean normalization.

Intersection: Solution

The visual elements guide understanding the geometric perspective of the solutions, where different colored lines represent equations, and the intersection indicates the equality of the solutions.

While Cramer's Rule provides formulaic solutions, alternative visual representations add cognitive engagement for conceptual depth.


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Cramer's Rule

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