Determinants

The determinant is a single number computed from a square matrix that encodes critical information about the system of equations the matrix represents. If the determinant is nonzero, the system has a unique solution. If the determinant is zero, the matrix is singular — the system either has no solution or infinitely many solutions.

Determinants appear throughout mathematics, physics, and engineering: they measure how a linear transformation scales area or volume, they determine whether a system of equations is solvable, and they are the foundation for Cramer’s Rule and matrix inverses.

The 2x2 Determinant

For a $2 \times 2$ matrix, the determinant is computed by the “cross-multiply and subtract” formula:

$\det \begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc$

The determinant is often written with vertical bars instead of brackets:

$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$

Pattern: Multiply the main diagonal ($a \cdot d$) and subtract the product of the off-diagonal ($b \cdot c$).

Example 1: Computing a 2x2 Determinant

$\begin{vmatrix} 3 & 7 \\ 2 & 5 \end{vmatrix} = 3(5) - 7(2) = 15 - 14 = 1$

Since the determinant is 1 (nonzero), this matrix is invertible and the corresponding system has a unique solution.

Example 2: A Singular 2x2 Matrix

$\begin{vmatrix} 4 & 6 \\ 2 & 3 \end{vmatrix} = 4(3) - 6(2) = 12 - 12 = 0$

The determinant is 0. This matrix is singular — it has no inverse, and the corresponding system either has no solution or infinitely many solutions. Notice that Row 2 is exactly half of Row 1: $[2 \; 3] = \frac{1}{2}[4 \; 6]$. This proportionality is what causes the zero determinant.

The 3x3 Determinant

For larger matrices, we use cofactor expansion (also called expansion by minors). You can expand along any row or column — the result is the same.

Cofactor Expansion Along the First Row

For a $3 \times 3$ matrix:

$\det \begin{bmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{bmatrix} = a_1 \begin{vmatrix} b_2 & b_3 \\ c_2 & c_3 \end{vmatrix} - a_2 \begin{vmatrix} b_1 & b_3 \\ c_1 & c_3 \end{vmatrix} + a_3 \begin{vmatrix} b_1 & b_2 \\ c_1 & c_2 \end{vmatrix}$

Each $2 \times 2$ determinant is called a minor — it is the determinant of the submatrix obtained by deleting the row and column of the entry being expanded. The cofactor is the minor multiplied by the appropriate sign.

The Sign Pattern (Checkerboard)

The signs in cofactor expansion alternate in a checkerboard pattern:

$\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$

This pattern determines whether each cofactor is added or subtracted.

Example 3: Computing a 3x3 Determinant

Compute $\det(A)$ where:

$A = \begin{bmatrix} 2 & 1 & -3 \\ 0 & 4 & 5 \\ 1 & -2 & 3 \end{bmatrix}$

Expand along the first row:

$\det(A) = 2 \begin{vmatrix} 4 & 5 \\ -2 & 3 \end{vmatrix} - 1 \begin{vmatrix} 0 & 5 \\ 1 & 3 \end{vmatrix} + (-3) \begin{vmatrix} 0 & 4 \\ 1 & -2 \end{vmatrix}$

Compute each $2 \times 2$ determinant:

$\begin{vmatrix} 4 & 5 \\ -2 & 3 \end{vmatrix} = 4(3) - 5(-2) = 12 + 10 = 22$

$\begin{vmatrix} 0 & 5 \\ 1 & 3 \end{vmatrix} = 0(3) - 5(1) = 0 - 5 = -5$

$\begin{vmatrix} 0 & 4 \\ 1 & -2 \end{vmatrix} = 0(-2) - 4(1) = 0 - 4 = -4$

Combine:

$\det(A) = 2(22) - 1(-5) + (-3)(-4) = 44 + 5 + 12 = 61$

Answer: $\det(A) = 61$

Example 4: Expanding Along a Different Row

You can expand along any row or column. Expanding along a row or column with zeros saves computation because those terms drop out.

Using the same matrix, expand along the first column (which contains a zero):

$\det(A) = 2 \begin{vmatrix} 4 & 5 \\ -2 & 3 \end{vmatrix} - 0 \begin{vmatrix} 1 & -3 \\ -2 & 3 \end{vmatrix} + 1 \begin{vmatrix} 1 & -3 \\ 4 & 5 \end{vmatrix}$

The zero in position $(2,1)$ eliminates the middle term entirely:

$\det(A) = 2(22) - 0 + 1\big(1(5) - (-3)(4)\big) = 44 + (5 + 12) = 44 + 17 = 61$

Same answer, but the zero saved us one $2 \times 2$ determinant computation.

Example 5: A Singular 3x3 Matrix

$B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

Expand along the first row:

$\det(B) = 1 \begin{vmatrix} 5 & 6 \\ 8 & 9 \end{vmatrix} - 2 \begin{vmatrix} 4 & 6 \\ 7 & 9 \end{vmatrix} + 3 \begin{vmatrix} 4 & 5 \\ 7 & 8 \end{vmatrix}$

$= 1(45 - 48) - 2(36 - 42) + 3(32 - 35)$

$= 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0$

The determinant is 0, so this matrix is singular. In fact, Row 3 equals $2 \times \text{Row 2} - \text{Row 1}$: $[7, 8, 9] = 2[4, 5, 6] - [1, 2, 3]$. Whenever one row is a linear combination of the others, the determinant is zero.

Properties of Determinants

These properties help you compute determinants more efficiently and understand what operations preserve (or change) the determinant.

The third property is particularly useful: the row replacement operation used in Gaussian elimination does not change the determinant. So you can reduce a matrix to triangular form and read off the determinant as the product of the diagonal entries.

Example 6: Determinant via Row Reduction

For matrix $A = \begin{bmatrix} 2 & 1 & -3 \\ 0 & 4 & 5 \\ 1 & -2 & 3 \end{bmatrix}$, swap $R_1$ and $R_3$ (this multiplies the determinant by $-1$):

$\begin{bmatrix} 1 & -2 & 3 \\ 0 & 4 & 5 \\ 2 & 1 & -3 \end{bmatrix}$

$R_3 - 2R_1 \to R_3$:

$\begin{bmatrix} 1 & -2 & 3 \\ 0 & 4 & 5 \\ 0 & 5 & -9 \end{bmatrix}$

$R_3 - \frac{5}{4}R_2 \to R_3$:

$\begin{bmatrix} 1 & -2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & -\frac{61}{4} \end{bmatrix}$

Product of diagonal: $1 \times 4 \times \left(-\frac{61}{4}\right) = -61$. We performed one row swap, so multiply by $-1$: $\det(A) = -(-61) = 61$. This matches our earlier answer.

What Does the Determinant Mean?

The determinant has a geometric interpretation: for a $2 \times 2$ matrix, $|\det(A)|$ is the area of the parallelogram formed by the column vectors. For a $3 \times 3$ matrix, $|\det(A)|$ is the volume of the parallelepiped formed by the column vectors.

• Nonzero determinant: The column vectors point in independent directions. The system has a unique solution, and the matrix is invertible.
• Zero determinant: The column vectors are coplanar (or collinear in 2D). The matrix is singular, the system does not have a unique solution, and no inverse exists.

Real-World Application: Electrical Network Solvability

When an electrician sets up Kirchhoff’s equations for a circuit with three loops, the coefficient matrix might be:

$A = \begin{bmatrix} 10 & -3 & 0 \\ -3 & 8 & -5 \\ 0 & -5 & 12 \end{bmatrix}$

Before attempting to solve, compute the determinant:

$\det(A) = 10(96 - 25) - (-3)(-36 - 0) + 0 = 10(71) + 3(-36) = 710 - 108 = 602$

Since $\det(A) = 602 \neq 0$, the circuit equations have a unique solution — the three loop currents are uniquely determined. If the determinant were zero, it would indicate a redundant equation (perhaps two loops are measuring the same constraint), and the system would need reformulation.

Common Mistakes

1. Wrong sign pattern. In cofactor expansion, the signs alternate starting with $+$ in position $(1,1)$. The sign of position $(i,j)$ is $(-1)^{i+j}$. Getting one sign wrong flips the answer.
2. Expanding and forgetting the entry. Each term in the expansion is the entry times the signed minor. Forgetting to multiply by the entry is a common error.
3. Confusing the minor’s rows and columns. When expanding along row 1 and computing the minor for $a_{12}$, delete row 1 and column 2 from the matrix. Be careful to delete the correct column.
4. Thinking determinant zero means “the matrix is zero.” A zero determinant means the matrix is singular (not invertible), not that the matrix itself is the zero matrix.

Practice Problems

Key Takeaways

• The 2x2 determinant is $ad - bc$ — multiply the diagonals and subtract
• The 3x3 determinant is computed by cofactor expansion: pick a row or column, multiply each entry by its signed minor, and add the results
• Signs follow a checkerboard pattern starting with $+$ at position $(1,1)$
• Expand along a row or column with zeros to minimize computation
• A zero determinant means the matrix is singular: no inverse exists, and the system does not have a unique solution
• A nonzero determinant means the matrix is invertible and the system has exactly one solution
• Row replacement operations do not change the determinant; row swaps multiply it by $-1$; scaling a row multiplies the determinant by the scalar

Return to College Algebra for more topics in this section.