Determinants

Jim Hefferon · GFDL + CC BY-SA 2.5 · Linear Algebra

The determinant is a single number that captures whether a square matrix is invertible and how it scales volume. det(A) = 0 means the map collapses a dimension. det(AB) = det(A) * det(B). Geometrically, the determinant of a 2-by-2 matrix is the signed area of the parallelogram spanned by its columns.

Cofactor expansion

The determinant of a 1-by-1 matrix is its single entry. For larger matrices, expand along any row or column: multiply each entry by its cofactor (the determinant of the submatrix with that row and column deleted, times a sign). The result is the same regardless of which row or column you pick.

Scheme

; Determinant of a 2x2 matrix: ad - bc

(define (det2 a b c d)
  (- (* a d) (* b c)))

(display "det((3,1),(1,2)) = ")
(display (det2 3 1 1 2))
(newline)
; 3*2 - 1*1 = 5

; Determinant of a 3x3 matrix by cofactor expansion along row 1
(define (det3 m)
  (let ((a (list-ref (list-ref m 0) 0))
        (b (list-ref (list-ref m 0) 1))
        (c (list-ref (list-ref m 0) 2))
        (d (list-ref (list-ref m 1) 0))
        (e (list-ref (list-ref m 1) 1))
        (f (list-ref (list-ref m 1) 2))
        (g (list-ref (list-ref m 2) 0))
        (h (list-ref (list-ref m 2) 1))
        (i (list-ref (list-ref m 2) 2)))
    (+ (* a (det2 e f h i))
       (- (* b (det2 d f g i)))
       (* c (det2 d e g h)))))

(define A '((1 2 3)
            (4 5 6)
            (7 8 9)))

(display "det(A) = ")
(display (det3 A))
; Should be 0 (rows are in arithmetic progression)

Properties of determinants

Three key properties. First, det(AB) = det(A) * det(B): the determinant is multiplicative. Second, row operations have predictable effects: swapping rows flips the sign, scaling a row multiplies the determinant, adding a multiple of one row to another leaves it unchanged. Third, det(A) is nonzero if and only if A is invertible.

Scheme

; Multiplicativity: det(AB) = det(A) * det(B)

(define (det2 a b c d) (- (* a d) (* b c)))

; A = ((2 1) (0 3)), det(A) = 6
; B = ((1 4) (2 5)), det(B) = -3
(define detA (det2 2 1 0 3))
(define detB (det2 1 4 2 5))

(display "det(A) = ") (display detA) (newline)
(display "det(B) = ") (display detB) (newline)
(display "det(A)*det(B) = ") (display (* detA detB)) (newline)

; Compute AB manually:
; AB = ((2*1+1*2, 2*4+1*5), (0*1+3*2, 0*4+3*5))
;    = ((4, 13), (6, 15))
(define detAB (det2 4 13 6 15))
(display "det(AB) = ") (display detAB) (newline)
; 4*15 - 13*6 = 60 - 78 = -18 = 6 * (-3)

(display "Equal? ") (display (= (* detA detB) detAB))

Row operations and determinants

Row operations are the algorithmic path to computing determinants. Reduce to echelon form, track the sign flips from row swaps, and multiply the diagonal entries. This is O(n^3), far better than cofactor expansion's O(n!).

Scheme

; Computing det via row reduction.
; A = ((2 6) (1 4))
; R2 <- R2 - (1/2)*R1: ((2 6) (0 1))
; det = product of pivots = 2 * 1 = 2

(define (det2 a b c d) (- (* a d) (* b c)))

; Direct computation:
(display "Direct: det = ")
(display (det2 2 6 1 4))
(newline)

; Via row reduction:
; After R2 <- R2 - (1/2)*R1, pivots are 2 and 1.
; No row swaps, so sign is +1.
(display "Via pivots: 2 * 1 = ")
(display (* 2 1))
(newline)

; With a row swap: swap R1 and R2
; ((1 4) (2 6)) — one swap, sign flips
; R2 <- R2 - 2*R1: ((1 4) (0 -2))
; det = (-1) * 1 * (-2) = 2
(display "With swap: (-1) * 1 * (-2) = ")
(display (* -1 1 -2))

Geometric interpretation

In 2D, |det(A)| is the area of the parallelogram spanned by the columns of A. In 3D, it is the volume of the parallelepiped. A negative determinant means the transformation reverses orientation. Zero determinant means the columns are coplanar, and the map squashes a dimension.

Scheme

; Signed area of a parallelogram
; Columns (2,0) and (0,3): axis-aligned rectangle, area = 6

(define (det2 a b c d) (- (* a d) (* b c)))

(display "Rectangle (2,0)x(0,3): area = ")
(display (det2 2 0 0 3))
(newline)

; Shear: columns (1,0) and (1,1)
; Parallelogram, area = 1 (shearing preserves area)
(display "Shear (1,0)x(1,1): area = ")
(display (det2 1 1 0 1))
(newline)

; Collapse: columns (1,2) and (2,4) — dependent
(display "Collapsed (1,2)x(2,4): area = ")
(display (det2 1 2 2 4))
(newline)
; Zero — the parallelogram degenerates to a line.

; Reflection flips sign: columns (0,1) and (1,0)
(display "Reflection (0,1)x(1,0): signed area = ")
(display (det2 0 1 1 0))

Notation reference

Symbol	Scheme	Python	Meaning
det(A)	(det2 a b c d)	np.linalg.det(A)	Determinant
det(AB) = det(A)det(B)	verified above	verified above	Multiplicativity
det(A) = 0	singular	singular	A is not invertible
cofactor Cij	(-1)^(i+j) * Mij	(-1)*(i+j) Mij	Signed minor
\|det(A)\|	(abs (det ...))	abs(det(A))	Volume scaling factor

Neighbors

Adjacent chapters

Ch.3 Maps Between Spaces — determinants measure properties of these maps
Ch.5 Similarity — eigenvalues are roots of det(A - lambda I) = 0

∫ Calculus Ch.11 Partial Derivatives — the Jacobian determinant for change of variables in multivariable integrals

Foundations (Wikipedia)

Translation notes

Hefferon's chapter also covers permutation-based definitions of the determinant, Cramer's rule, and the Laplace expansion in full generality. This page focuses on the 2-by-2 and 3-by-3 cases and the key properties. For the permutation approach and proofs of multiplicativity, see the original text.

Full chapter: Hefferon, Chapter Four — Determinants.

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