Determinants

  • A determinant is a numerical value that can be calculated from the elements of a square matrix.
  • It provides important information about the matrix and its properties.
  • Notation: The determinant of a matrix A is denoted as |A|.
  • The determinant of a 2x2 matrix is calculated as: |A| = ad - bc
    • where a, b, c, and d are the elements of the matrix.
  • The determinant of a 3x3 matrix is calculated as: |A| = a(ei - fh) - b(di - fg) + c(dh - eg)
    • where a, b, c, d, e, f, g, and h are the elements of the matrix.

Properties of Determinants

  • The determinant of a matrix with all elements zero is zero.
  • If two rows or columns of a matrix are interchanged, the sign of the determinant changes.
  • If any row or column of a matrix is multiplied by a scalar, the determinant is multiplied by that scalar.
  • If two rows or columns of a matrix are equal, the determinant is zero.
  • If two rows or columns of a matrix are proportional, the determinant is zero.
  • Example: Calculate the determinant of the matrix A = [\begin{pmatrix} a & b \ c & d \ \end{pmatrix}]
    • |A| = ad - bc

Solving Equations using Determinants

  • Determinants can be used to solve systems of linear equations.
  • Consider the system of equations: [\begin{align*} ax + by = p \ cx + dy = q \ \end{align*}]
  • If |A| ≠ 0, where A is the coefficient matrix, the system has a unique solution given by:
    • x = (\frac{{|A_x|}}{{|A|}}) and y = (\frac{{|A_y|}}{{|A|}})
      • Where Ax is formed by replacing the x-column in matrix A with the constants p and q.
      • Similarly, Ay is formed by replacing the y-column in matrix A with the constants p and q.
  • Example: Solve the system of equations using determinants: [\begin{align*} 3x + 2y = 7 \ 5x - 4y = -1 \ \end{align*}]

Cramer’s Rule

  • Cramer’s rule is used to solve systems of linear equations using determinants.
  • Consider the system of equations: [\begin{align*} ax + by + cz = p \ dx + ey + fz = q \ gx + hy + iz = r \ \end{align*}]
    • Where A is the coefficient matrix and X is the matrix of variables.
  • If |A| ≠ 0, the system has a unique solution given by:
    • x = (\frac{{|A_x|}}{{|A|}}), y = (\frac{{|A_y|}}{{|A|}}), and z = (\frac{{|A_z|}}{{|A|}})
      • Where Ax, Ay, and Az are formed by replacing the x, y, and z-columns in matrix A with the constants p, q, and r.
  • Example: Solve the system of equations using Cramer’s rule: [\begin{align*} x - 2y + 3z = 6 \ 2x + y + z = 2 \ 3x - y + 2z = -6 \ \end{align*}]

Adjoint and Inverse of a Matrix

  • The adjoint of a matrix is obtained by taking the transpose of the cofactor matrix.
  • The cofactor matrix is formed by taking the determinant of each element of the matrix without its row and column.
  • The inverse of a matrix A is denoted as A^(-1), and it is calculated as:
    • A^(-1) = (\frac{{\text{{adj}}(A)}}{{|A|}})
  • Example: Find the adjoint and inverse of the matrix A = [\begin{pmatrix}

3 & 2 \

1 & 4 \ \end{pmatrix}]

  • Solution:
    1. Find the cofactor matrix C.
    2. Transpose the cofactor matrix to obtain the adjoint matrix.
    3. Calculate the determinant |A|.
    4. Use the formula A^(-1) = (\frac{{\text{{adj}}(A)}}{{|A|}}) to find the inverse matrix.

Determinants - Solved Examples: System of Equations

  • Example 1: Solve the following system of equations using determinants: [\begin{align*} 2x + 3y - z &= 12 \ x - y + z &= 4 \ 3x + 2y + 4z &= 24 \ \end{align*}]
  • Solution:
    1. Set up the coefficient matrix A and the constant matrix B.
    2. Calculate the determinant |A|.
    3. Use the formula x = (\frac{{|A_x|}}{{|A|}}), y = (\frac{{|A_y|}}{{|A|}}), and z = (\frac{{|A_z|}}{{|A|}}) to find the solution.
  • Example 2: Solve the following system of equations using determinants: [\begin{align*} 3x - 2y + z &= 5 \ x + y - z &= -2 \ 2x - 3y + 4z &= 12 \ \end{align*}]
  • Solution:
    1. Set up the coefficient matrix A and the constant matrix B.
    2. Calculate the determinant |A|.
    3. Use the formula x = (\frac{{|A_x|}}{{|A|}}), y = (\frac{{|A_y|}}{{|A|}}), and z = (\frac{{|A_z|}}{{|A|}}) to find the solution.
  • Example 3: Solve the following system of equations using determinants: [\begin{align*} 4x - 2y + 3z &= 1 \ 6x + y - z &= 7 \ 2x - 3y + 2z &= 3 \ \end{align*}]
  • Solution:
    1. Set up the coefficient matrix A and the constant matrix B.
    2. Calculate the determinant |A|.
    3. Use the formula x = (\frac{{|A_x|}}{{|A|}}), y = (\frac{{|A_y|}}{{|A|}}), and z = (\frac{{|A_z|}}{{|A|}}) to find the solution.
  • Example 4: Solve the following system of equations using determinants: [\begin{align*} 5x + 4y - 2z &= 13 \ 3x - y + 5z &= 2 \ 6x + 2y + z &= 0 \ \end{align*}]
  • Solution:
    1. Set up the coefficient matrix A and the constant matrix B.
    2. Calculate the determinant |A|.
    3. Use the formula x = (\frac{{|A_x|}}{{|A|}}), y = (\frac{{|A_y|}}{{|A|}}), and z = (\frac{{|A_z|}}{{|A|}}) to find the solution.

Slide s 21-30:

Determinants - Solved Examples: System of equations

Slide 21:

  • Example 1: Solve the following system of equations using determinants:
    • 2x + 3y - z = 12
    • x - y + z = 4
    • 3x + 2y + 4z = 24

Slide 22:

  • Solution:
    1. Set up the coefficient matrix A and the constant matrix B.
    2. Calculate the determinant |A|.
    3. Use the formula x = |Ax|/|A|, y = |Ay|/|A|, and z = |Az|/|A| to find the solution.

Slide 23:

  • Example 2: Solve the following system of equations using determinants:
    • 3x - 2y + z = 5
    • x + y - z = -2
    • 2x - 3y + 4z = 12

Slide 24:

  • Solution:
    1. Set up the coefficient matrix A and the constant matrix B.
    2. Calculate the determinant |A|.
    3. Use the formula x = |Ax|/|A|, y = |Ay|/|A|, and z = |Az|/|A| to find the solution.

Slide 25:

  • Example 3: Solve the following system of equations using determinants:
    • 4x - 2y + 3z = 1
    • 6x + y - z = 7
    • 2x - 3y + 2z = 3

Slide 26:

  • Solution:
    1. Set up the coefficient matrix A and the constant matrix B.
    2. Calculate the determinant |A|.
    3. Use the formula x = |Ax|/|A|, y = |Ay|/|A|, and z = |Az|/|A| to find the solution.

Slide 27:

  • Example 4: Solve the following system of equations using determinants:
    • 5x + 4y - 2z = 13
    • 3x - y + 5z = 2
    • 6x + 2y + z = 0

Slide 28:

  • Solution:
    1. Set up the coefficient matrix A and the constant matrix B.
    2. Calculate the determinant |A|.
    3. Use the formula x = |Ax|/|A|, y = |Ay|/|A|, and z = |Az|/|A| to find the solution.

Slide 29:

  • Summary:
    • Determinants can be used to solve systems of linear equations.
    • By using determinants, we can find the unique solution for a system of equations.
    • Cramer’s rule provides a method to solve systems of equations using determinants.

Slide 30:

  • Practice:
    • Solve the following system of equations using determinants:
      • 2x + 4y - 3z = 10
      • 3x - 2y + z = 4
      • x + y + 2z = 8
    • Challenge:
      • Solve the following system of equations using determinants:
        • 4x + 2y - 3z = 14
        • 2x - y + 3z = 3
        • 3x + 4y + 2z = 12