Section outline

  • COURSE SYNOPSIS

    COURSE SYNOPSIS

       

    Definition of matrices: addition and scalar multiplication, matrix multiplication, transpose of matrix, symmetric matrix, skewed symmetric matrix, diagonal matrix and identity matrix. Elementary row operation, reduced row-echelon form, singular matrix and nonsingular matrix, inverse of a matrix. 


    Systems of linear equations: solving by matrix method, homogeneous systems of linear equations, geometrical representation, solutions to AX=B with two unknowns.

    Determinant of a square matrix, cofactor, finding determinant by using cofactors, properties of determinant and adjoint of a matrix. Relationship between adjoint and determinant of square matrices. Finding inverses of matrices using determinants, Cramer's rule, eigenvalues and eigenvectors.

    Vector spaces, subspaces, basis and dimension, addition of spaces, isomorphic vector spaces.


    Linear transformation, definition and it's properties. Matrix representation of a linear transformation, kernel and range, change of basis. Inner product. Gram-Schmidt process and projection. Innerproduct space, orthorgonal set and orthonormal basis.

     

    Get YouTube Transcripts

    Please watch the Essence of Linear Algebra (est. time 5mins)



  • Systems of Linear Equations and Matrices




  • Summary of Subtopic and SLT




    Systems of Linear Equations and Matrices
    No Sub Topic Student Learning Time (SLT)
     1 Systems of Linear Equations 3
     2 Matrices
    		 3
     3 Matrix Multiplication 3
     4  Algebraic Properties of Matrix Operations 3
     5 Special Types of Matrices and Partitioned Matrices 3
     6  Matrix Transformation
    		 3



  • Introduction

    If we are required to arrange or organize things in an orderly manner, how will we do it?  Most of us will spontaneously line up or place objects that will form rows and columns.  Observe how students are usually placed in a class or examination hall, participants are arranged in a procession, and so on.  When we arrange something in the form of rows and columns, we form what is known in mathematics as a matrix.    

    There are many reasons or benefits when we arrange things in the form of columns and rows. One of them is that we will be able to identify/ recognize/ describe an object by stating the rows and columns where an object is located.  With this,  identification can be made accurately, quickly, and clearly.  In this chapter, we will look at what a matrix is and some types of matrices as well as the properties of matrices.  We will also study some operations on matrices and the properties of those operations.  



  • Instructional Materials





  • Self-learning Activities


    Learning Activities
    Activity Instructions
    Learning Activity 1.1 Please watch this video to look at examples on different types of matrices
    Source: youtube
     

    Get YouTube Transcripts
    Learning Activity 1.2

    Please attempt all the questions below. Solutions will be discussed in WEBEX #1
     Learning Activity 1.3  

    Please watch this video to look at matrix multiplication and associated properties.

    Source: youtube
     Learning Activity 1.4
     Please watch this video to look at different examples on Matrix Operations

    Source: youtube
     Learning Activity 1.5  

    Please watch this video to understand partitioned matrices better.

    Source: youtube
     Learning Activity 1.6  

    Please watch this video to look at different examples on Matrix Transformations, reflections and rotations.

    Source: youtube



  • Self Assessment


    Questions Learning Outcomes


    Matrix Recognition and Notation

    Question: Recognizing Matrix and Understanding Terms

    Question:

    Consider the following matrix A:

    A = \begin{bmatrix} 3 & -1 & 2 \\ 0 & 4 & 7 \\ -2 & 6 & 1 \end{bmatrix}

    1. Identify the order (size) of matrix A.
    2. Define and identify the elements of matrix A.
    3. What is the main diagonal of matrix A?
    4. Determine the transpose of matrix A.
    5. Write the row and column vectors associated with matrix A.
    6. Recognize the special types of matrix A, if any.

    Solution:

    1. Order (Size) of Matrix A:

      The order of matrix A is 3 \times 3 (3 rows and 3 columns).

    2. Elements of Matrix A:

      The elements of matrix A are the individual numbers within the matrix. A_{1,1} = 3, \quad A_{1,2} = -1, \quad A_{1,3} = 2 A_{2,1} = 0, \quad A_{2,2} = 4, \quad A_{2,3} = 7 A_{3,1} = -2, \quad A_{3,2} = 6, \quad A_{3,3} = 1

    3. Main Diagonal of Matrix A:

      The main diagonal is the set of elements where the row index equals the column index. Main diagonal: 3, 4, 1

    4. Transpose of Matrix A:

      The transpose of matrix A, denoted as A^T, is obtained by swapping its rows and columns. A^T = \begin{bmatrix} 3 & 0 & -2 \\ -1 & 4 & 6 \\ 2 & 7 & 1 \end{bmatrix}

    5. Row and Column Vectors:

      Row vector associated with matrix A: \mathbf{r}_1 = [3, -1, 2]

      Column vector associated with matrix A: \mathbf{c}_1 = \begin{bmatrix} 3 \\ 0 \\ -2 \end{bmatrix}

    6. Special Types of Matrix:

      A is a square matrix (number of rows = number of columns).

      A is a symmetric matrix (equal to its transpose).

    These assessments demonstrates how to recognize a matrix, identify its order, elements, main diagonal, transpose, row and column vectors, and recognize any special types associated with the matrix.

    		  Recognize matrix, know the terms associated with matrix and its notation
     System of Linear Equations

    Question 1: Gaussian Elimination Method

    Solve the system of equations:

    2x + y - z = 8 -3x - y + 2z = -11 -2x + y + 2z = -3

    Step 1: Write the augmented matrix.

    \left[\begin{array}{ccc|c} 2 & 1 & -1 & 8 \\ -3 & -1 & 2 & -11 \\ -2 & 1 & 2 & -3 \end{array}\right]

    Step 2: Perform row operations to get an upper triangular form.

    \left[\begin{array}{ccc|c} 2 & 1 & -1 & 8 \\ 0 & 1 & 4 & 5 \\ 0 & 3 & 0 & 5 \end{array}\right]

    Step 3: Back-substitute to find the solutions.

    z = 1 y + 4z = 5 \implies y = 1 2x + y - z = 8 \implies x = 2

    So, the solution is x = 2, y = 1, z = 1 .


    Question 2: Gauss-Jordan Elimination Method

    Solve the system of equations:

    3x - y = 9 2x + y = 4

    Step 1: Write the augmented matrix.

    \left[\begin{array}{cc|c} 3 & -1 & 9 \\ 2 & 1 & 4 \end{array}\right]

    Step 2: Perform row operations to get reduced row-echelon form.

    \left[\begin{array}{cc|c} 1 & 0 & 3 \\ 0 & 1 & 2 \end{array}\right]

    Step 3: Interpret the solutions.

    x = 3 y = 2

    So, the solution is x = 3, y = 2 .


    These assessments illustrate the process of determining the size of a matrix and finding its transpose.

    		 Determine the size and transpose of a matrix 

     Matrix Types and Properties

    Example: Recognizing Types of Matrices

    Question:

    Consider the matrix C:

    C = \begin{bmatrix} 2 & 0 & 0 \\ -1 & 3 & 0 \\ 4 & 1 & -2 \end{bmatrix}

    1. Determine the type of matrix C (triangular, diagonal, identity, zero, symmetric).
    2. If D is a matrix of the same order as C given by D = \begin{bmatrix} 1 & -1 & 4 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \end{bmatrix}, find the product CD and identify its type.

    Solution:

    1. Determine the Type of Matrix C:

      • Triangular Matrix:

        C is an upper triangular matrix because all entries below the main diagonal are zero.

      • Diagonal Matrix:

        C is not a diagonal matrix because not all off-diagonal entries are zero.

      • Identity Matrix:

        C is not an identity matrix because the main diagonal entries are not all ones.

      • Zero Matrix:

        C is not a zero matrix because it has non-zero entries.

      • Symmetric Matrix:

        C is not a symmetric matrix because it is not equal to its transpose.

    2. Product of C and D:

      3. Let's find the product CD.

      CD = \begin{bmatrix} 2 & 0 & 0 \\ -1 & 3 & 0 \\ 4 & 1 & -2 \end{bmatrix} \begin{bmatrix} 1 & -1 & 4 \\ 0 & 2 & 1 \\ 0 & 0 & 3 \end{bmatrix}

      Calculate the product:

      CD = \begin{bmatrix} 2 & -2 & 8 \\ -1 & 6 & 3 \\ 4 & 1 & -6 \end{bmatrix}

      Determine the Type of Matrix CD:

      • Triangular Matrix:

        CD is also an upper triangular matrix because all entries below the main diagonal are zero.

      • Diagonal Matrix:

        CD is not a diagonal matrix because not all off-diagonal entries are zero.

      • Identity Matrix:

        CD is not an identity matrix because the main diagonal entries are not all ones.

      • Zero Matrix:

        CD is not a zero matrix because it has non-zero entries.

      • Symmetric Matrix:

        CD is not a symmetric matrix because it is not equal to its transpose.

    This assessment demonstrates the process of recognizing different types of matrices and determining their properties.

    		 Recognize a few types of matrices such as triangular, diagonal, identity, zero, and symmetric matrices

    Algebraic Properties of Matrix Operations

    Questions and Solutions: Algebraic Properties of Matrix Operations

    Question 1:

    Consider the matrices A, B, and C:

    A = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix} B = \begin{bmatrix} 1 & 0 \\ -2 & 5 \end{bmatrix} C = \begin{bmatrix} 0 & 3 \\ 1 & 2 \end{bmatrix}

    1. Find A + B and B + A. Verify if matrix addition is commutative.
    2. Calculate A + (B + C) and (A + B) + C. Verify if matrix addition is associative.
    3. Determine 2A and A + A. Verify if scalar multiplication and matrix addition satisfy the distributive property.

    Solution 1:

    1. Matrix Addition Commutativity:
      • A + B = \begin{bmatrix} 3 & -1 \\ 1 & 9 \end{bmatrix}
      • B + A = \begin{bmatrix} 3 & -1 \\ 1 & 9 \end{bmatrix}
      • Since A + B = B + A, matrix addition is commutative.
    2. Matrix Addition Associativity:
      • A + (B + C) = \begin{bmatrix} 3 & 2 \\ 2 & 10 \end{bmatrix}
      • (A + B) + C = \begin{bmatrix} 3 & 2 \\ 2 & 10 \end{bmatrix}
      • Since A + (B + C) = (A + B) + C, matrix addition is associative.
    3. Scalar Multiplication and Distributive Property:
      • 2A = \begin{bmatrix} 4 & -2 \\ 6 & 8 \end{bmatrix}
      • A + A = \begin{bmatrix} 4 & -2 \\ 6 & 8 \end{bmatrix}
      • 2A and A + A satisfy the distributive property.

    Question 2:

    Now, consider the matrices X, Y, and Z:

    X = \begin{bmatrix} 1 & 2 \\ 0 & -3 \end{bmatrix} Y = \begin{bmatrix} -1 & 4 \\ 2 & 1 \end{bmatrix} Z = \begin{bmatrix} 3 & 1 \\ -2 & 0 \end{bmatrix}

    1. Compute X \cdot Y and Y \cdot X. Verify if matrix multiplication is commutative.
    2. Evaluate (XY)Z and X(YZ). Verify if matrix multiplication is associative.

    Solution 2:

    1. Matrix Multiplication Commutativity:
      • X \cdot Y = \begin{bmatrix} 5 & 9 \\ -6 & -12 \end{bmatrix}
      • Y \cdot X = \begin{bmatrix} -3 & 0 \\ 5 & -6 \end{bmatrix}
      • Since X \cdot Y \neq Y \cdot X, matrix multiplication is not commutative.
    2. Matrix Multiplication Associativity:
      • (XY)Z = \begin{bmatrix} -27 & 9 \\ -24 & 12 \end{bmatrix}
      • X(YZ) = \begin{bmatrix} -27 & 9 \\ -24 & 12 \end{bmatrix}
      • Matrix multiplication is associative.


    This assessment demonstrates the process of determination of algebraic properties for matrix operations, including commutativity, associativity, and the distributive property.

    		 Determine the Algebraic Properties of Matrix Operations


     


  • Summary

     

     


    In conclusion, linear algebra, encompassing systems of linear equations and matrices, has proven to be an essential tool across different disciplines. By developing a thorough understanding of these concepts, you will be equipped to tackle real-life problems efficiently and make informed decisions. Remember, practice is the key to mastering linear algebra. Now, let's dive into our fascinating journey through the world of systems of linear equations and matrices.

      

     


  • Keywords

    1. Linear algebra: A branch of mathematics that deals with systems of linear equations and matrices.
    2. Systems of linear equations: A set of two or more linear equations with the same variables.
    3. Matrices: Rectangular arrays of numbers.
    4. Matrix notation: A way to represent matrices using brackets.
    5. Graphical method: A way to solve systems of linear equations by graphing their corresponding lines.
    6. Substitution method: A method for solving systems of linear equations by replacing one variable with an expression.
    7. Elimination method: A method for solving systems of linear equations by manipulating the equations to eliminate a variable.
    8. Matrix addition: The process of adding corresponding elements of two matrices to obtain a new matrix.
    9. Matrix subtraction: The process of subtracting corresponding elements of two matrices to obtain a new matrix.
    10. Scalar multiplication: The process of multiplying a matrix by a scalar value.
    11. Matrix multiplication: The process of multiplying two matrices to obtain a new matrix.
    12. Square matrix: A matrix with an equal number of rows and columns.
    13. Identity matrix: A square matrix with ones on the main diagonal and zeros elsewhere.
    14. Rectangular matrix: A matrix with a different number of rows and columns.
    15. Real-life applications: The practical uses of systems of linear equations and matrices in various fields such as engineering, physics, economics, and computer science.



  • References

    1. "Linear Algebra and Its Applications" by David C. Lay, Steven R. Lay, and Judi J. McDonald:

      • This is a widely used and well-regarded textbook that covers the essentials of linear algebra and its applications. It provides clear explanations and many practical examples.


    2. "Linear Algebra Done Right" by Sheldon Axler:

      • This book takes a more theoretical approach to linear algebra and is known for its clarity and emphasis on vector spaces. It's a great choice for students who want a deeper understanding of the subject.

    3. "Introduction to Linear Algebra" by Gilbert Strang:

      • Written by a renowned mathematician, this book is suitable for both introductory and advanced courses in linear algebra. It covers a wide range of topics and includes practical applications.