4.2 Determinants and Cramer's Rule

Determinants of Square Matrices

Determinants and Cramer's Rule connect matrices to systems of equations and geometric transformations. A determinant boils a square matrix down to a single number, and that number reveals whether the matrix is invertible, how it scales area or volume, and whether it flips orientation. Cramer's Rule then puts determinants to work by using them to solve systems of linear equations directly.

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Calculating Determinants

The determinant is a scalar value computed from the elements of a square matrix. You'll see it written as $det(A)$ or $|A|$.

For a 2×2 matrix, the formula is straightforward: multiply the main diagonal and subtract the product of the off-diagonal.

$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \quad det(A) = ad - bc$

For example, if $A = \begin{bmatrix} 3 & 5 \\ 2 & 7 \end{bmatrix}$, then $det(A) = (3)(7) - (5)(2) = 21 - 10 = 11$.

For a 3×3 matrix, you use cofactor expansion (also called expansion along a row or column). Here's how it works along the first row:

1. Pick the first row element, multiply it by the determinant of the 2×2 matrix you get by deleting that element's row and column.
2. Move to the next element in the row and do the same, but flip the sign.
3. Continue alternating signs across the row: $+, -, +$.

$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$

Two quick base cases:

• The determinant of a 1×1 matrix is just the value of its single element.
• The determinant of a triangular matrix (upper or lower) is the product of its diagonal entries. For example, $\begin{bmatrix} 2 & 5 & 1 \\ 0 & 3 & 4 \\ 0 & 0 & 6 \end{bmatrix}$ has determinant $2 \cdot 3 \cdot 6 = 36$.

Special Cases That Give a Zero Determinant

A few matrix patterns guarantee $det(A) = 0$:

• Any row or column is all zeros.
• Two rows (or two columns) are identical.
• One row is a scalar multiple of another row (the rows are linearly dependent).

Recognizing these patterns can save you a lot of computation.

Properties of Determinants

These properties are useful both for simplifying determinant calculations and for understanding how matrix operations affect the determinant.

Transpose and Determinant

The determinant doesn't change when you transpose a matrix:

$det(A) = det(A^T)$

For example, $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ has $det(A) = 4 - 6 = -2$. Its transpose $A^T = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$ also has $det(A^T) = 4 - 6 = -2$. This means every property that applies to rows also applies to columns.

Row and Column Operations

• Swapping two rows (or columns) flips the sign of the determinant.
• Multiplying a single row (or column) by a scalar $k$ multiplies the determinant by $k$.
• Adding a multiple of one row to another row does not change the determinant.

Be careful with the scaling rule: multiplying one row by $k$ multiplies the determinant by $k$. If you multiply the entire $n \times n$ matrix by $k$, the determinant gets multiplied by $k^n$.

Matrix Products and Invertibility

$det(AB) = det(A) \times det(B)$

This is a clean, powerful result. For example, if $det(A) = -2$ and $det(B) = -2$, then $det(AB) = (-2)(-2) = 4$.

The determinant also tells you whether a matrix is invertible (non-singular):

• $det(A) \neq 0$: the matrix is invertible.
• $det(A) = 0$: the matrix is singular and has no inverse.

This connects directly to systems of equations: a coefficient matrix with a nonzero determinant means the system has exactly one solution.

Cramer's Rule for Systems

Solving Linear Systems

Cramer's Rule gives you a formula for each variable in a system of $n$ equations with $n$ unknowns, using only determinants. The idea: replace one column of the coefficient matrix at a time with the constants from the right-hand side, then divide that new determinant by the original.

For a 2×2 system:

$\begin{cases} ax + by = e \\ cx + dy = f \end{cases}$

Let $D = det\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ (the coefficient matrix determinant). Then:

$x = \frac{det\begin{bmatrix} e & b \\ f & d \end{bmatrix}}{D}, \quad y = \frac{det\begin{bmatrix} a & e \\ c & f \end{bmatrix}}{D}$

Worked example: Solve $\begin{cases} 2x + 3y = 8 \\ x - 4y = -5 \end{cases}$

1. Compute the coefficient determinant: $D = (2)(-4) - (3)(1) = -8 - 3 = -11$

2. For $x$, replace the first column with the constants: $D_x = det\begin{bmatrix} 8 & 3 \\ -5 & -4 \end{bmatrix} = (8)(-4) - (3)(-5) = -32 + 15 = -17$

3. For $y$, replace the second column with the constants: $D_y = det\begin{bmatrix} 2 & 8 \\ 1 & -5 \end{bmatrix} = (2)(-5) - (8)(1) = -10 - 8 = -18$

4. Divide: $x = \frac{-17}{-11} = \frac{17}{11}, \quad y = \frac{-18}{-11} = \frac{18}{11}$

The general rule for variable $x_i$: replace the $i$-th column of the coefficient matrix with the column of constants, compute that determinant, and divide by $D$.

Applicability and Efficiency

Cramer's Rule only works when the system has a unique solution, meaning $D \neq 0$. If $D = 0$, the system is either inconsistent (no solution) or dependent (infinitely many solutions), and Cramer's Rule can't distinguish between the two.

For 2×2 and 3×3 systems, Cramer's Rule is practical and often faster than other methods. For larger systems, it becomes computationally expensive: a 3×3 system requires four 3×3 determinants, and the workload grows rapidly from there. Methods like Gaussian elimination are more efficient for bigger systems.

Geometric Interpretation of Determinants

Area and Volume

The absolute value of a determinant has a direct geometric meaning:

• For a 2×2 matrix, $|det(A)|$ equals the area of the parallelogram formed by the column vectors.
• For a 3×3 matrix, $|det(A)|$ equals the volume of the parallelepiped (a 3D parallelogram) formed by the column vectors.

For example, the matrix $A = \begin{bmatrix} 3 & 0 \\ 0 & 2 \end{bmatrix}$ has $|det(A)| = 6$. The column vectors $(3, 0)$ and $(0, 2)$ form a rectangle with area 6, which checks out.

A determinant of zero means the vectors are linearly dependent, so they don't span a full parallelogram (in 2D) or parallelepiped (in 3D). The "shape" collapses to a lower dimension.

Orientation and Invertibility

The sign of the determinant tells you about orientation:

• Positive determinant: the transformation preserves orientation (no "flipping").
• Negative determinant: the transformation reverses orientation (like a reflection).

If $det(A) = 0$, the matrix is not invertible, and the transformation it represents squashes space down to a lower dimension. For instance, $A = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$ has $det(A) = 0$. The second column is just twice the first, so this matrix maps the entire 2D plane onto a single line. All the area gets crushed to zero, which is exactly what the zero determinant tells you.