What Is a Determinant?
The determinant of a square matrix $A$ — written $\det(A)$ or $|A|$ — is a single scalar computed from the matrix's entries.
It carries multiple meanings:
Algebraically. A polynomial in the matrix entries that's zero exactly when the matrix is singular (non-invertible).
Geometrically. The scaling factor for areas (in 2D) or volumes (in higher dimensions) under the linear transformation the matrix represents.
Practically. A test for whether a system of linear equations has a unique solution ($\det \neq 0$) or not ($\det = 0$).
Determinant of a 2×2 Matrix
For $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$:
$$\det(A) = ad - bc$$
Memory pattern. Multiply the main diagonal, multiply the off-diagonal, subtract.
Worked example. Find $\det\begin{pmatrix} 5 & 3 \ 2 & 4 \end{pmatrix}$.
$\det = (5)(4) - (3)(2) = 20 - 6 = 14$.
Determinant of a 3×3 Matrix
For $A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{pmatrix}$, two methods.
Method 1: Cofactor Expansion (Along the First Row)
$$\det(A) = a_{11}\det\begin{pmatrix} a_{22} & a_{23} \ a_{32} & a_{33} \end{pmatrix} - a_{12}\det\begin{pmatrix} a_{21} & a_{23} \ a_{31} & a_{33} \end{pmatrix} + a_{13}\det\begin{pmatrix} a_{21} & a_{22} \ a_{31} & a_{32} \end{pmatrix}$$
Apply alternating signs $+, -, +, \ldots$ and use the $2 \times 2$ determinant formula on each submatrix.
Method 2: Sarrus' Rule (3×3 Only)
Write the matrix and copy the first two columns to the right:
$$\begin{pmatrix} a_{11} & a_{12} & a_{13} & | & a_{11} & a_{12} \ a_{21} & a_{22} & a_{23} & | & a_{21} & a_{22} \ a_{31} & a_{32} & a_{33} & | & a_{31} & a_{32} \end{pmatrix}$$
Sum the products of three diagonals going down-right, subtract the sum of three diagonals going down-left.
Sarrus only works for $3 \times 3$. For $4 \times 4$ and larger, use cofactor expansion (or row reduction).
Three Worked Examples — Quick, Standard, Stretch
Quick — 2×2
$\det\begin{pmatrix} 7 & 2 \ 3 & 4 \end{pmatrix} = 7(4) - 2(3) = 28 - 6 = 22$.
Standard — 3×3 by Cofactor Expansion
$\det\begin{pmatrix} 2 & 1 & 3 \ 0 & 4 & 5 \ 1 & 0 & 6 \end{pmatrix}$.
Expand along the first row:
$$= 2 \det\begin{pmatrix} 4 & 5 \ 0 & 6 \end{pmatrix} - 1 \det\begin{pmatrix} 0 & 5 \ 1 & 6 \end{pmatrix} + 3 \det\begin{pmatrix} 0 & 4 \ 1 & 0 \end{pmatrix}$$
$$= 2(24 - 0) - 1(0 - 5) + 3(0 - 4) = 48 + 5 - 12 = 41$$
Stretch — Use Properties
Find $\det\begin{pmatrix} 0 & 0 & 5 \ 0 & 3 & 0 \ 7 & 0 & 0 \end{pmatrix}$.
Method: column expansion or recognise as a permutation matrix with scaling.
Expand along column 1: only $a_{31} = 7$ is nonzero.
$$\det = 7 \cdot (-1)^{3+1} \det\begin{pmatrix} 0 & 5 \ 3 & 0 \end{pmatrix} = 7 \cdot 1 \cdot (0 - 15) = -105$$
Properties of the Determinant
A handful of properties make calculations dramatically faster — and reveal what the determinant means.
$\det(I) = 1$ — the identity matrix.
$\det(AB) = \det(A) \cdot \det(B)$ — multiplicative property.
$\det(A^T) = \det(A)$ — determinant of the transpose equals the original.
$\det(A^{-1}) = \dfrac{1}{\det(A)}$ — inverse scales by the reciprocal.
$\det(kA) = k^n \det(A)$ — for an $n \times n$ matrix scaled by $k$.
Swapping rows changes the sign of the determinant.
Adding a multiple of one row to another doesn't change the determinant.
If any row (or column) is all zeros, $\det(A) = 0$.
If two rows (or columns) are equal, $\det(A) = 0$.
For triangular matrices (upper or lower), $\det = $ product of diagonal entries.
The last property is the basis of the fastest practical determinant algorithm: row-reduce to upper triangular, then multiply the diagonal.
Why Does the Determinant Matter? (The Real-World GROUND)
"A determinant is a single number that tells you everything the matrix does to space." — adapted geometric intuition.
The determinant appears anywhere matrices appear:
Solving linear systems. $A\vec{x} = \vec{b}$ has a unique solution exactly when $\det(A) \neq 0$. Cramer's rule expresses each unknown as a ratio of determinants.
Eigenvalue computation. The eigenvalues of a matrix $A$ are the roots of the characteristic polynomial $\det(A - \lambda I) = 0$.
Change of variables in integration. The Jacobian determinant tells you how volumes transform under a coordinate change — essential in multivariable calculus.
Computer graphics. The determinant of a transformation matrix tells you whether the transformation preserves orientation (positive) or flips it (negative).
Physics — quantum mechanics. Slater determinants describe antisymmetric multi-electron wavefunctions.
Cryptography. Some cipher algorithms require the encryption matrix to have determinant coprime to the alphabet size.
The determinant was introduced in essentially its modern form by Gabriel Cramer in 1750 (Cramer's rule) and Pierre-Simon Laplace in 1772 (cofactor expansion).
Learn more: Inverse of 2x2 Matrix
A Worked Example — Wrong Path First
Find $\det\begin{pmatrix} 1 & 2 \ 2 & 4 \end{pmatrix}$.
The intuitive (wrong) approach. A student computes $1 \cdot 4 + 2 \cdot 2 = 8$.
Why it fails. The student added instead of subtracting. The determinant formula is $ad - bc$, not $ad + bc$.
The correct method. $\det = (1)(4) - (2)(2) = 4 - 4 = 0$.
Interpretation. Determinant zero means the matrix is singular — and indeed, the rows are proportional (row 2 = $2 \times$ row 1). The matrix collapses 2D space to a 1D line.
What Are the Most Common Mistakes With Determinants?
Mistake 1: Sign errors in cofactor expansion
Where it slips in: Forgetting the alternating sign pattern.
The fix: $+, -, +, -, \ldots$ starting with $+$ in position $(1,1)$. Write the signs over the matrix before expanding.
Mistake 2: Applying Sarrus' rule to a 4×4 matrix
Where it slips in: Sarrus' diagonal trick only works for $3 \times 3$.
The fix: For $4 \times 4$ and larger, use cofactor expansion or row reduction. There's no "Sarrus for 4×4."
Mistake 3: Confusing $\det(A + B)$ with $\det(A) + \det(B)$
Where it slips in: Treating the determinant as linear in $A$.
The fix: $\det(A + B) \neq \det(A) + \det(B)$ in general. The determinant is multilinear in rows (or columns), not linear in the whole matrix. Be careful.
Key Takeaways
The determinant is a single number computed from a square matrix — it captures area-scaling, invertibility, and orientation.
2×2 formula: $ad - bc$. 3×3: cofactor expansion or Sarrus' rule.
Zero determinant ⇔ singular matrix ⇔ no inverse ⇔ collapsed transformation.
Multiplicative property: $\det(AB) = \det(A) \cdot \det(B)$.
Row operations matter: swapping rows flips the sign; scaling a row scales the determinant; adding multiples of one row to another doesn't change it.
A Practical Next Step
Try these three before moving on to eigenvalues.
Compute $\det\begin{pmatrix} 4 & 3 \ 2 & 5 \end{pmatrix}$.
Compute $\det\begin{pmatrix} 1 & 0 & 2 \ 3 & 4 & 5 \ 0 & 1 & 6 \end{pmatrix}$.
Show that $\det\begin{pmatrix} 2 & 4 \ 3 & 6 \end{pmatrix} = 0$ and explain why.
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