Overview
The determinant is a scalar value computed from a square matrix that provides important information about the matrix, including whether it is invertible and the volume scaling factor of the associated linear transformation.
Definition for 2×2 Matrices
For a 2 × 2 2 \times 2 2 × 2
A = [ a b c d ] A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} A = [ a c b d ]
det ( A ) = a d − b c \det(A) = ad - bc det ( A ) = a d − b c
Example
A = [ 3 2 1 4 ] A = \begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix} A = [ 3 1 2 4 ]
det ( A ) = 3 ( 4 ) − 2 ( 1 ) = 12 − 2 = 10 \det(A) = 3(4) - 2(1) = 12 - 2 = 10 det ( A ) = 3 ( 4 ) − 2 ( 1 ) = 12 − 2 = 10
Definition for 3×3 Matrices
For a 3 × 3 3 \times 3 3 × 3
A = [ a b c d e f g h i ] A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} A = a d g b e h c f i
det ( A ) = a ( e i − f h ) − b ( d i − f g ) + c ( d h − e g ) \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg) det ( A ) = a ( e i − f h ) − b ( d i − f g ) + c ( d h − e g )
Sarrus' Rule (3×3 only)
det ( A ) = a e i + b f g + c d h − c e g − b d i − a f h \det(A) = aei + bfg + cdh - ceg - bdi - afh det ( A ) = a e i + b f g + c d h − ce g − b d i − a f h
Example
A = [ 1 2 3 4 5 6 7 8 9 ] A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} A = 1 4 7 2 5 8 3 6 9
det ( A ) = 1 ( 45 − 48 ) − 2 ( 36 − 42 ) + 3 ( 32 − 35 ) = 1 ( − 3 ) − 2 ( − 6 ) + 3 ( − 3 ) = − 3 + 12 − 9 = 0 \det(A) = 1(45-48) - 2(36-42) + 3(32-35) = 1(-3) - 2(-6) + 3(-3) = -3 + 12 - 9 = 0 det ( A ) = 1 ( 45 − 48 ) − 2 ( 36 − 42 ) + 3 ( 32 − 35 ) = 1 ( − 3 ) − 2 ( − 6 ) + 3 ( − 3 ) = − 3 + 12 − 9 = 0
Notation
Common notations for determinant:
det ( A ) = ∣ a b c d ∣ \det(A) = \begin{vmatrix} a & b \\ c & d \end{vmatrix} det ( A ) = a c b d
Geometric Interpretation
In 2D
∣ det ( A ) ∣ \lvert\det(A)\rvert ∣ det ( A )∣
In 3D
∣ det ( A ) ∣ \lvert\det(A)\rvert ∣ det ( A )∣
Sign
Positive: preserves orientation
Negative: reverses orientation
Zero: collapses dimension (area/volume = 0)
Key Values
Determinant Meaning det ( A ) ≠ 0 \det(A) \neq 0 det ( A ) = 0 A A A det ( A ) = 0 \det(A) = 0 det ( A ) = 0 A A A det ( I ) = 1 \det(I) = 1 det ( I ) = 1 Identity matrix det ( O ) = 0 \det(O) = 0 det ( O ) = 0 Zero matrix
Determinant and Linear Independence
For n n n R n \mathbb{R}^n R n A A A
det ( A ) ≠ 0 ⇔ \det(A) \neq 0 \Leftrightarrow det ( A ) = 0 ⇔ det ( A ) = 0 ⇔ \det(A) = 0 \Leftrightarrow det ( A ) = 0 ⇔
Examples
Example 1: Checking Invertibility
A = [ 1 2 3 6 ] A = \begin{bmatrix} 1 & 2 \\ 3 & 6 \end{bmatrix} A = [ 1 3 2 6 ]
det ( A ) = 1 ( 6 ) − 2 ( 3 ) = 6 − 6 = 0 \det(A) = 1(6) - 2(3) = 6 - 6 = 0 det ( A ) = 1 ( 6 ) − 2 ( 3 ) = 6 − 6 = 0
A A A
Example 2: Area Calculation
Parallelogram with vertices at origin, ( 2 , 1 ) (2,1) ( 2 , 1 ) ( 1 , 3 ) (1,3) ( 1 , 3 ) ( 3 , 4 ) (3,4) ( 3 , 4 )
Column vectors: [ 2 1 ] \begin{bmatrix} 2 \\ 1 \end{bmatrix} [ 2 1 ] [ 1 3 ] \begin{bmatrix} 1 \\ 3 \end{bmatrix} [ 1 3 ]
Area = ∣ det ( 2 1 1 3 ) ∣ = ∣ 2 ( 3 ) − 1 ( 1 ) ∣ = ∣ 5 ∣ = 5 \text{Area} = \left\lvert \det\begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} \right\rvert = \lvert 2(3) - 1(1) \rvert = \lvert 5 \rvert = 5 Area = det ( 2 1 1 3 ) = ∣ 2 ( 3 ) − 1 ( 1 )∣ = ∣ 5 ∣ = 5
Example 3: 3×3 Determinant
A = [ 2 0 1 3 1 2 1 0 2 ] A = \begin{bmatrix} 2 & 0 & 1 \\ 3 & 1 & 2 \\ 1 & 0 & 2 \end{bmatrix} A = 2 3 1 0 1 0 1 2 2
det ( A ) = 2 ( 1 × 2 − 2 × 0 ) − 0 ( 3 × 2 − 2 × 1 ) + 1 ( 3 × 0 − 1 × 1 ) = 2 ( 2 ) − 0 + 1 ( − 1 ) = 4 − 1 = 3 \det(A) = 2(1 \times 2 - 2 \times 0) - 0(3 \times 2 - 2 \times 1) + 1(3 \times 0 - 1 \times 1) = 2(2) - 0 + 1(-1) = 4 - 1 = 3 det ( A ) = 2 ( 1 × 2 − 2 × 0 ) − 0 ( 3 × 2 − 2 × 1 ) + 1 ( 3 × 0 − 1 × 1 ) = 2 ( 2 ) − 0 + 1 ( − 1 ) = 4 − 1 = 3