Study Notes on Cramer's Rule and Systems of Equations

Cramer's Rule and Systems of Equations

Definition of a Matrix

• A matrix is defined as a rectangular arrangement of numbers arranged in rows and columns.

• Conceptually, a matrix can be understood as an "array" of numbers.

Definition of a Determinant

• The determinant is a value that is associated with a square matrix in linear algebra.

  • It can be computed using specific arithmetic expressions based on the entries of the matrix.

  • Additionally, there exist other methods to determine the value of the determinant.

Determinant of a 2x2 Matrix

• To find the determinant of a 2x2 matrix, represented as:
  $\begin{bmatrix} a & b \ c & d \end{bmatrix}$

• The formula for the determinant is:
  $D = (a \cdot d) - (b \cdot c)$

Evaluating Determinants

• The session includes practical exercises labeled "You Try," where users are encouraged to evaluate the determinant of provided matrices.

• Users are initiatively prompted to evaluate each determinant from accompanying examples.

Application of Cramer's Rule to Solve Systems of Equations

• Cramer's Rule is deployed for solving systems of equations represented in matrix form:

  • Given two equations, in standard form as:

  • $-5x - 5y = 25$

  • $-2x - 4y = 16$

• The general steps to apply Cramer's Rule include:

  1. Ensure that both equations are in standard form.

  2. Identify and name the coefficients:

  • $x<em>1, x</em>2, y<em>1, y</em>2, c<em>1, c</em>2$

  1. Set up the matrix incorporating x's, y's, and c's:

  • Matrix formation includes the coefficients of the variables and the constants.

  1. Construct the determinants for:

  • $D$ (the main determinant),

  • $D_x$ (determinant for variable x),

  • $D_y$ (determinant for variable y).

Detailed Derivation of Determinants

• For the determinants:

  • $D$ is derived from the matrix with x and y coefficients.

  • $D_x$ is formed by replacing the column of x coefficients with constant terms (c).

  • $D_y$ is formed by replacing the column of y coefficients with the constant terms (c).

Example Problems

• Following the theory, several examples are provided which demonstrate the application of Cramer's Rule:

  1. Example 1:

  • System of Equations:

    • $4x + 2y = 25$

    • $3 + y = x$

    • $-2x + y = 3$

    • $-4x + 2y = 8$

  1. Example 2:

  • System of Equations:

    • $5x - 4 = 2y$

    • $8 + 4y = x$

  1. Example 3:

  • System of Equations:

    • $x - 4y = -4$

    • $-4x + 16y = 16$

Homework Assignment

• Homework is assigned to reinforce learning:

  • Task: Complete problems from the back page of the worksheet titled "Solving System of Equations with Cramer's Rule."

  • Specifically, problems 1-10 are designated as mandatory for completion.

Conclusion

• The notes provide an exhaustive overview of Cramer's Rule, emphasizing matrix representation, determinants, and systematic equation solving, alongside practice and homework references.