Solving Systems with Cramer’s Rule: Fresh Take

  • Evaluate 2 × 2 and 3 × 3 determinants
  • Use Cramer’s Rule to solve a system of equations in two variables
  • Use Cramer’s Rule to solve a system of three equations in three variables

Evaluate 2 × 2 and 3 × 3 Determinants

The Main Idea

  • Determinant:
    • A real number calculated from a square matrix
    • Notation: [latex]\det(A)[/latex] or [latex]|A|[/latex]
    • Used to determine if a matrix is invertible
  • 2 × 2 Determinant Formula:
    • For [latex]A = \left[\begin{array}{cc}a & b\\ c & d\end{array}\right][/latex]
    • [latex]\det(A) = ad - bc[/latex]
  • 3 x 3 Determinants: use calculator function det()

Key Techniques

  1. For 2 × 2 Matrices:
    • Multiply diagonal entries (top-left to bottom-right)
    • Subtract product of other diagonal (top-right to bottom-left)
  2. For 3 × 3 Matrices:
    • Augment matrix by repeating first two columns
    • Add products of three diagonals going down-right
    • Subtract products of three diagonals going up-right
Find the determinant of [latex]A = \left[\begin{array}{cc}5 & 2\\ -6 & 3\end{array}\right][/latex].

You can view the transcript for “Finding the determinant of a 2×2 matrix | Matrices | Precalculus | Khan Academy” here (opens in new window).

Find the determinant of [latex]A = \left[\begin{array}{ccc}0 & 2 & 1\\ 3 & -1 & 1\\ 4 & 0 & 1\end{array}\right][/latex].

You can view the transcript for “Determinant of a 3x3 matrix using Augmented Matrices” here (opens in new window).

Use Cramer's Rule to Solve Systems in Two Variables

The Main Idea

  • Cramer's Rule:
    • Method using determinants to solve systems
    • Works when number of equations equals number of variables
    • If [latex]D = 0[/latex], system has no solution or infinite solutions
  • For system [latex]\begin{array}{l}{a}_{1}x+{b}_{1}y={c}_{1}\\ {a}_{2}x+{b}_{2}y={c}_{2}\end{array}[/latex]:
    • [latex]D = \left\rvert\begin{array}{cc}{a}_{1} & {b}_{1}\\ {a}_{2} & {b}_{2}\end{array}\right\rvert[/latex] (coefficient matrix)
    • [latex]D_x = \left\rvert\begin{array}{cc}{c}_{1} & {b}_{1}\\ {c}_{2} & {b}_{2}\end{array}\right\rvert[/latex] (replace x-column with constants)
    • [latex]D_y = \left\rvert\begin{array}{cc}{a}_{1} & {c}_{1}\\ {a}_{2} & {c}_{2}\end{array}\right\rvert[/latex] (replace y-column with constants)
    • [latex]x = \frac{D_x}{D}[/latex], [latex]y = \frac{D_y}{D}[/latex]

Key Techniques

  1. Find [latex]D[/latex] (determinant of coefficient matrix)
  2. Find [latex]D_x[/latex] (replace x-column with constant column)
  3. Find [latex]D_y[/latex] (replace y-column with constant column)
  4. Calculate [latex]x = \frac{D_x}{D}[/latex] and [latex]y = \frac{D_y}{D}[/latex]
Solve using Cramer's Rule: [latex]\begin{array}{l}12x + 3y = 15\\ 2x - 3y = 13\end{array}[/latex]

You can view the transcript for “Learn How to Use Cramer's Rule to Solve a 2 x 2 System of Equations” here (opens in new window).

Use Cramer's Rule to Solve Systems in Three Variables

The Main Idea

  • For 3 × 3 Systems:
    • Same pattern as 2 × 2 systems
    • Replace each variable's column with constants to find that variable
    • [latex]x = \frac{D_x}{D}[/latex], [latex]y = \frac{D_y}{D}[/latex], [latex]z = \frac{D_z}{D}[/latex]

Key Techniques

  1. Find [latex]D[/latex] (determinant of coefficient matrix)
  2. Find [latex]D_x[/latex], [latex]D_y[/latex], and [latex]D_z[/latex] by replacing appropriate columns
  3. Calculate each variable using the formulas
Solve using Cramer's Rule:
[latex]\begin{array}{l}x + y - z = 6\\ 3x - 2y + z = -5\\ x + 3y - 2z = 14\end{array}[/latex]

You can view the transcript for “How do you use Cramer’s Rule to solve Systems of 3 Linear Equations? The Easiest Method!” here (opens in new window).