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 ↓


Matrix Operations


Matrices are everywhere in mathematics and are particularly useful in many fields such as physics, computer science and economics. In Grade 11 Mathematics, you begin to delve deeper into these fascinating structures, understanding how they are applied and manipulated through various operations. This page will guide you through the fundamentals of matrix operations in an accessible and comprehensive way.

What is a matrix?

A matrix is essentially a rectangular array of numbers arranged in rows and columns. These numbers are called the elements of the matrix. The size or dimension of a matrix is defined by the number of rows and columns it contains. For example, a matrix with 3 rows and 2 columns would be a 3x2 matrix.

Here's an example:

A = | 1 2 | | 3 4 | | 5 6 |

Here, matrix A is a 3x2 matrix.

Types of matrices

Before delving deeper into the operations, it is important to understand the different types of matrices:

  • Row Matrix: A matrix with only one row. Example:
    R = | 7 8 9 |
  • Column Matrix: A matrix with only one column. Example:
    C = | 1 | | 4 | | 7 |
  • Square matrix: A matrix with the same number of rows and columns. Example:
    Q = | 5 6 | | 7 8 |
  • Diagonal Matrix: A square matrix where all the elements outside the main diagonal are zero. Example:
    D = | 3 0 0 | | 0 5 0 | | 0 0 9 |
  • Identity Matrix: A square matrix where all the elements of the main diagonal are ones, and all other elements are zero. Usually denoted by I Example:
    I = | 1 0 0 | | 0 1 0 | | 0 0 1 |

Addition and subtraction of matrices

Addition and subtraction of matrices are relatively simple operations, but there is an important condition: the matrices to be added or subtracted must be of the same dimension.

Matrix addition

To add two matrices, simply add their corresponding elements. For example:

A = | 2 3 | B = | 4 1 | | 5 7 | | 0 2 | A + B = | (2+4) (3+1) | | (5+0) (7+2) | = | 6 4 | | 5 9 |

Matrix subtraction

To subtract one matrix from another, subtract their corresponding elements. For example:

A = | 6 4 | B = | 1 3 | | 2 5 | | 4 7 | A - B = | (6-1) (4-3) | | (2-4) (5-7) | = | 5 1 | |-2 -2 |

Scalar multiplication

Scalar multiplication involves multiplying each element of a matrix by a scalar (real number). For example:

k = 3, A = | 2 1 | | 3 4 | kA = | 3*2 3*1 | | 3*3 3*4 | = | 6 3 | | 9 12 |

Matrix multiplication

Matrix multiplication is a little more complicated than addition, subtraction, or scalar multiplication. Two matrices can be multiplied only if the number of columns in the first matrix is equal to the number of rows in the second matrix.

For example, consider two matrices A and B where:

A = | 3 4 2 | a 1x3 matrix B = | 13 9 7 15| b 3x4 matrix | 8 7 4 6 | | 6 4 0 3 | Resultant Matrix C, where C = A x B, would be of size 1x4.

To calculate the element in the ith row and jth column of a matrix product A x B , perform the dot product of the ith row of A and the jth column of B For the first element you do it like this:

C11 = (3 * 13) + (4 * 8) + (2 * 6) = 39 + 32 + 12 = 83

Compute all the elements in this manner to obtain the resultant matrix C

Transposing a matrix

The transpose of a matrix is formed by swapping its rows with its columns and vice versa. The transpose of a matrix A is usually denoted as A T

For example, if:

A = | 3 7 | | 5 6 | | 8 9 |

Then, the transpose of A is:

A T = | 3 5 8 | | 7 6 9 |

Identity matrix and inverse matrix

The identity matrix for multiplication is like the number 1 for multiplying the numbers. The identity matrix I is a square matrix in which all diagonal elements are equal to one and other elements are equal to zero.

Inverse matrix

Only square matrices have an inverse, and not all square matrices have an inverse. If it exists, then the inverse of the matrix A is the matrix A -1 , such that:

A * A -1 = I

where I is the identity matrix.

Conclusion

Understanding matrix operations is a fundamental aspect of dealing with matrices. These operations form the backbone for more complex mathematical functions and are essential tools in many scientific analyses.

Feel encouraged to practice with different matrices to solidify your understanding of these operations. Over time, the concepts will become intuitive as you become more comfortable with the math associated with matrices.


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