Matrix and Determinant - Question 1

  • Topic: System of Linear Equations
  • Objective: To solve a system of linear equations using matrix and determinant method.

Definition of a System of Linear Equations

  • Definition: A system of linear equations is a set of equations where the unknowns appear in a linear form.
  • Example:
    • (2x + 3y = 7)
    • (4x - y = -1)

Matrix Representation

  • We can represent a system of linear equations in matrix form.
  • Example:
    • The above system of equations can be written as: [ \begin{pmatrix} 2 & 3 \ 4 & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 7 \ -1 \end{pmatrix} ]

Solving a System of Linear Equations Using Matrix and Determinant Method

  • Steps to solve a system of linear equations using the matrix and determinant method:
    1. Write the given system of equations in matrix form.
    2. Calculate the determinant of the coefficient matrix.
    3. If the determinant is non-zero, find the inverses of the coefficient matrix.
    4. Multiply the inverse of the coefficient matrix with the column matrix of constants.
    5. The resulting matrix will give the values of the unknown variables.

Determinant of a Matrix

  • Definition: The determinant of a square matrix can be calculated using a specific formula.
  • Example:
    • For a 2x2 matrix (A = \begin{pmatrix} a & b \ c & d \end{pmatrix}), the determinant is given by (|A| = ad - bc).

Inverse of a Matrix

  • Definition: The inverse of a square matrix can be calculated using the formula.
  • Example:
    • For a 2x2 matrix (A = \begin{pmatrix} a & b \ c & d \end{pmatrix}), the inverse is given by: [A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \ -c & a \end{pmatrix}] Sure! Here are slides 11 to 20 on the topic “Matrix and Determinant - Question 1 (On System of Linear Equations)”.

Slide 11

  • Question: Solve the following system of linear equations:
    • (3x - 2y = 4)
    • (2x + y = 6)

Slide 12

  • Step 1: Write the given system of equations in matrix form:
    • (\begin{pmatrix} 3 & -2 \ 2 & 1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 4 \ 6 \end{pmatrix})

Slide 13

  • Step 2: Calculate the determinant of the coefficient matrix:
    • (\begin{vmatrix} 3 & -2 \ 2 & 1 \end{vmatrix} = (3 \times 1) - (-2 \times 2) = 3 + 4 = 7)

Slide 14

  • Step 3: Since the determinant is non-zero, we can find the inverses of the coefficient matrix:
    • (\begin{pmatrix} 3 & -2 \ 2 & 1 \end{pmatrix}^{-1} = \frac{1}{7} \begin{pmatrix} 1 & 2 \ -2 & 3 \end{pmatrix})

Slide 15

  • Step 4: Multiply the inverse of the coefficient matrix with the column matrix of constants:
    • (\frac{1}{7} \begin{pmatrix} 1 & 2 \ -2 & 3 \end{pmatrix} \begin{pmatrix} 4 \ 6 \end{pmatrix} = \begin{pmatrix} \frac{22}{7} \ \frac{6}{7} \end{pmatrix})

Slide 16

  • Step 5: The resulting matrix gives the values of the unknown variables:
    • (x = \frac{22}{7})
    • (y = \frac{6}{7})

Slide 17

  • Therefore, the solution to the given system of linear equations is:
    • (x = \frac{22}{7})
    • (y = \frac{6}{7})

Slide 18

  • Example: Solve the system of equations using matrix and determinant method:
    • (5x + 3y = 16)
    • (2x - y = 4)

Slide 19

  • Step 1: Write the given system of equations in matrix form:
    • (\begin{pmatrix} 5 & 3 \ 2 & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 16 \ 4 \end{pmatrix})

Slide 20

  • Step 2: Calculate the determinant of the coefficient matrix:
    • (\begin{vmatrix} 5 & 3 \ 2 & -1 \end{vmatrix} = (5 \times -1) - (3 \times 2) = -5 - 6 = -11) Hope these slides help in your lecture! Sure! Here are slides 21 to 30 on the topic “Matrix and Determinant - Question 1 (On System of Linear Equations)”.

Slide 21

  • Example (continued): Solve the system of equations using matrix and determinant method:
    • (\begin{pmatrix} 5 & 3 \ 2 & -1 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix} = \begin{pmatrix} 16 \ 4 \end{pmatrix})

Slide 22

  • Step 3: Since the determinant is non-zero, we can find the inverses of the coefficient matrix:
    • (\begin{pmatrix} 5 & 3 \ 2 & -1 \end{pmatrix}^{-1} = \frac{1}{-11} \begin{pmatrix} -1 & -3 \ -2 & 5 \end{pmatrix})

Slide 23

  • Step 4: Multiply the inverse of the coefficient matrix with the column matrix of constants:
    • (\frac{1}{-11} \begin{pmatrix} -1 & -3 \ -2 & 5 \end{pmatrix} \begin{pmatrix} 16 \ 4 \end{pmatrix} = \begin{pmatrix} -4 \ 12 \end{pmatrix})

Slide 24

  • Step 5: The resulting matrix gives the values of the unknown variables:
    • (x = -4)
    • (y = 12)

Slide 25

  • Therefore, the solution to the given system of linear equations is:
    • (x = -4)
    • (y = 12)

Slide 26

  • Summary:
    • A system of linear equations can be represented using matrices.
    • The determinant of the coefficient matrix is used to determine if a unique solution exists.
    • If the determinant is non-zero, the system has a unique solution.
    • The inverse of the coefficient matrix is used to find the values of the unknown variables.

Slide 27

  • Practice Problem 1: Solve the following system of linear equations using matrix and determinant method:
    • (3x - y = 5)
    • (2x + 4y = -6)

Slide 28

  • Practice Problem 2: Solve the following system of linear equations using matrix and determinant method:
    • (5x + 2y = -1)
    • (-3x + 6y = 4)

Slide 29

  • Practice Problem 3: Solve the following system of linear equations using matrix and determinant method:
    • (4x - 3y = 2)
    • (-x + 2y = 3)

Slide 30

  • Thank you for attending the lecture!
  • Practice solving more problems to strengthen your understanding of matrix and determinant method.
  • Feel free to ask any questions you may have.