9.8 Solving Systems with Cramer's Rule

Determinants and Their Properties

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Calculation of matrix determinants

The determinant is a single number you can compute from any square matrix. It tells you whether the matrix is invertible and shows up directly in Cramer's Rule, so you need to be comfortable calculating it.

2×2 determinant:

For the matrix $\begin{bmatrix}a & b \\ c & d\end{bmatrix}$, the determinant is:

$ad - bc$

Multiply the main diagonal, then subtract the product of the off-diagonal. That's it.

3×3 determinant:

For the matrix $\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$, the determinant is:

$a(ei - fh) - b(di - fg) + c(dh - eg)$

This is called cofactor expansion along the first row. Here's how to do it step by step:

1. Take the first element $a$, cross out its row and column, and find the 2×2 determinant of what's left: $ei - fh$. Multiply by $a$.

2. Take the second element $b$, cross out its row and column, and find the 2×2 determinant: $di - fg$. Multiply by $-b$ (note the sign change).

3. Take the third element $c$, cross out its row and column, and find the 2×2 determinant: $dh - eg$. Multiply by $c$.

4. Add the three results together.

The sign pattern along the first row alternates: $+, -, +$. This pattern extends for larger matrices too.

You may also see the Rule of Sarrus for 3×3 matrices: copy the first two columns to the right of the matrix, then add the products of the three downward diagonals and subtract the products of the three upward diagonals. It gives the same answer but only works for 3×3.

Properties of determinants

These properties are worth knowing because they can save you computation time and help you check your work:

• Row/column swap: Interchanging any two rows (or columns) flips the sign of the determinant.
• Scalar multiplication: Multiplying a single row or column by $k$ multiplies the determinant by $k$.
• Row addition: Adding a multiple of one row to another row does not change the determinant.
• Transpose: The determinant of a matrix equals the determinant of its transpose.
• Product rule: $\det(AB) = \det(A) \cdot \det(B)$.
• Invertibility: A matrix is invertible if and only if its determinant is non-zero. If the determinant is zero, the matrix is singular.

Solving Systems with Cramer's Rule

Cramer's Rule gives you a direct formula for each variable in a system of linear equations, expressed entirely in terms of determinants. The core idea: replace one column of the coefficient matrix with the constants from the right-hand side, take that determinant, and divide by the determinant of the original coefficient matrix.

Application of Cramer's Rule

For a 2×2 system $\begin{cases}a_1x + b_1y = c_1 \\ a_2x + b_2y = c_2\end{cases}$:

Let $D = \begin{vmatrix}a_1 & b_1 \\ a_2 & b_2\end{vmatrix}$ (the determinant of the coefficient matrix).

$x = \frac{\begin{vmatrix}c_1 & b_1 \\ c_2 & b_2\end{vmatrix}}{D}$ and $y = \frac{\begin{vmatrix}a_1 & c_1 \\ a_2 & c_2\end{vmatrix}}{D}$

Notice the pattern: to solve for $x$, replace the x-column (first column) with the constants. To solve for $y$, replace the y-column (second column) with the constants. The denominator is always $D$.

For a 3×3 system $\begin{cases}a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3\end{cases}$:

The same pattern holds. Let $D = \begin{vmatrix}a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3\end{vmatrix}$.

$x = \frac{\begin{vmatrix}d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3\end{vmatrix}}{D}$, $y = \frac{\begin{vmatrix}a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3\end{vmatrix}}{D}$, and $z = \frac{\begin{vmatrix}a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3\end{vmatrix}}{D}$

Step-by-step process for any system:

1. Write the coefficient matrix and compute its determinant $D$.
2. Check that $D \neq 0$. If $D = 0$, Cramer's Rule doesn't apply (the system is either inconsistent or has infinitely many solutions).
3. For each variable, form a new matrix by replacing that variable's column in the coefficient matrix with the column of constants.
4. Compute the determinant of each new matrix.
5. Divide each new determinant by $D$ to get the value of that variable.

Quick example (2×2): Solve $\begin{cases}3x + 2y = 7 \\ x - y = 1\end{cases}$

• $D = \begin{vmatrix}3 & 2 \\ 1 & -1\end{vmatrix} = (3)(-1) - (2)(1) = -5$
• $x = \frac{\begin{vmatrix}7 & 2 \\ 1 & -1\end{vmatrix}}{-5} = \frac{(7)(-1) - (2)(1)}{-5} = \frac{-9}{-5} = \frac{9}{5}$
• $y = \frac{\begin{vmatrix}3 & 7 \\ 1 & 1\end{vmatrix}}{-5} = \frac{(3)(1) - (7)(1)}{-5} = \frac{-4}{-5} = \frac{4}{5}$

Cramer's Rule vs other methods

• Cramer's Rule works well for 2×2 and 3×3 systems, especially on exams where you need a quick, formulaic approach.
• For larger systems, the number of determinant calculations grows very fast (factorially), making Gaussian elimination or matrix inversion far more practical.
• One nice advantage: if you only need the value of one variable, you can compute just two determinants instead of solving the entire system.

Real-world interpretation of solutions

The solution to a system represents the set of values that satisfy all equations at once. Depending on context, these values might represent:

• Intersection points of lines or planes in geometry
• Equilibrium forces or velocities in physics
• Prices or quantities that balance supply and demand in economics

Systems of Equations and Their Properties

Types of systems

• Consistent system: Has at least one solution. If there's exactly one solution, it's called independent. If there are infinitely many, it's dependent.
• Inconsistent system: Has no solution. Graphically, this means the lines (or planes) never all intersect at a common point.

For Cramer's Rule specifically: the system has a unique solution when $D \neq 0$. When $D = 0$, you can't use Cramer's Rule, and you'll need another method to determine whether the system is inconsistent or dependent.

Matrix representation

Any system of linear equations can be written in matrix form as $AX = B$, where:

• $A$ is the coefficient matrix (the grid of coefficients from the left-hand side)
• $X$ is the column vector of variables
• $B$ is the column vector of constants from the right-hand side

The augmented matrix $[A | B]$ combines both into one matrix, which is what you'd use for Gaussian elimination. For Cramer's Rule, you work with $A$ and $B$ separately.