Determinants
1. What is a Determinant? - Number from Square Matrix
Determinant is a single number you get from square matrix, written with vertical bars |A| not brackets. For 1×1 [a] determinant is a. For 2×2 [[a,b],[c,d]] determinant is ad-bc, cross multiply subtract. Example [[1,2],[3,4]] det=1×4-2×3=4-6=-2. For 3×3 you expand along row. Determinant tells if matrix invertible: if det≠0 invertible, if det=0 not invertible (singular).
Determinant |A| scalar from square matrix. 1×1 det=a, 2×2 det=ad-bc. 3×3 expand. |A|≠0 invertible, |A|=0 singular non-invertible. Notation vertical bars.
• Determinant |A| is number from square matrix, vertical bars
• 1×1 [a] → a, 2×2 [[a,b],[c,d]] → ad-bc
• Example [[1,2],[3,4]]=1×4-2×3=-2
• |A|≠0 invertible, |A|=0 singular
• Only square matrices have determinant
2. Expansion of Determinant - How to Calculate 3×3
To find 3×3 determinant, pick a row or column and expand using minors and cofactors. Example row 1: det = a11×C11 + a12×C12 + a13×C13 where C is cofactor (-1)^(i+j) × minor. Shortcut Sarrus rule for 3×3: repeat first two columns after matrix, sum of diagonal products minus sum of other diagonal products. Better to expand along row with zeros to make calculation easy.
Expand along any row/col: |A|=Σ a_ij C_ij where C_ij=(-1)^(i+j) M_ij. Choose row with max zeros. Sarrus only for 3×3: copy first 2 cols, sum \\ diagonals - sum // diagonals. Cofactor expansion works for any order.
• Expand: |A|=a11C11+a12C12+a13C13 along row 1
• C_ij=(-1)^(i+j) M_ij minor
• Choose row/col with zeros for easy calculation
• Sarrus rule only 3×3: copy first 2 cols, \\ minus //
• Cofactor sign: + - + / - + - / + - + pattern
3. Properties of Determinants - 7 Rules That Save Time
Property 1: |A^T|=|A| transpose same determinant. Property 2: Swapping two rows/cols changes sign. Property 3: If two rows/cols identical, det=0. Property 4: Multiplying a row/col by k multiplies det by k, so |kA|=k^n|A| for n×n. Property 5: If row is sum of two rows, det splits sum. Property 6: Adding multiple of one row to another doesn't change det (R_i→R_i+kR_j). Property 7: If all elements of row/col zero, det=0. Use these to make zeros then expand.
Properties: |A^T|=|A|, R_i↔R_j ⇒ det→-det, identical rows det=0, kR_i ⇒ det→k det, |kA|=k^n|A|, row sum splits, R_i→R_i+kR_j det unchanged, zero row det=0. Use to create zeros.
• |A^T|=|A| transpose same
• Swap two rows: det changes sign, identical rows det=0
• Multiply row by k: det becomes k×det, |kA|=k^n|A|
• Row sum: det splits into sum
• R_i→R_i+kR_j det unchanged (most useful)
• Zero row/col → det=0
• Use properties to make zeros then expand
4. Minors and Cofactors - Building Blocks
Minor M_ij is determinant after deleting row i and column j. Example matrix [[1,2,3],[4,5,6],[7,8,9]] minor M_11 is det[[5,6],[8,9]]=5×9-6×8=-3. Cofactor C_ij=(-1)^(i+j) M_ij, sign depends on position. Sign matrix: + - + / - + - / + - + for 3×3. Cofactors used to find adjoint and inverse. Minor is small determinant, cofactor is signed minor.
Minor M_ij = det after delete R_i C_j. Cofactor C_ij=(-1)^(i+j) M_ij. Sign matrix + - + / - + - / + - + . Cofactor matrix = matrix of C_ij. Minors used for expansion.
• Minor M_ij = det of matrix after deleting row i col j
• Example: Delete row1 col1, remaining 2×2 det is minor
• Cofactor C_ij=(-1)^(i+j) M_ij signed
• Sign pattern: + - + / - + - / + - +
• Cofactor matrix = all C_ij
• Used for adjoint and inverse
5. Adjoint of a Matrix - Transpose of Cofactor
Adjoint adj A is transpose of cofactor matrix. Steps: Find all cofactors C_ij, make cofactor matrix C=[C_ij], then transpose it → adj A = C^T. Example 2×2 [[a,b],[c,d]] cofactor [[d,-c],[-b,a]] transpose [[d,-b],[-c,a]] so adj = [[d,-b],[-c,a]]. For 3×3 do same. Property: A(adj A)=(adj A)A=|A|I. If |A|≠0, inverse exists.
adj A = (Cofactor matrix)^T = transpose of C_ij. For 2×2 [[a,b],[c,d]] adj=[[d,-b],[-c,a]]. Property A adj A = adj A A = |A| I_n. If |A|≠0 then A⁻¹ = adj A / |A|.
• Steps: 1) Find all cofactors C_ij 2) Make cofactor matrix 3) Transpose → adj A
• 2×2 [[a,b],[c,d]] adj=[[d,-b],[-c,a]] swap diagonal change sign off-diagonal
• Property: A adj A = adj A A = |A| I
• If |A|=0 adjoint still exists but inverse not
• Adjoint is bridge to inverse
6. Inverse of Matrix - Using Adjoint
If |A|≠0, inverse A⁻¹ = (1/|A|) adj A. Example A=[[1,2],[3,4]] det=-2 adj=[[4,-2],[-3,1]] inverse=(-1/2)[[4,-2],[-3,1]]=[[-2,1],[1.5,-0.5]]. Check A A⁻¹=I. Properties: (A⁻¹)⁻¹=A, (AB)⁻¹=B⁻¹A⁻¹ reverse, (A^T)⁻¹=(A⁻¹)^T, |A⁻¹|=1/|A|. If |A|=0, inverse does not exist, matrix singular. Use inverse to solve AX=B → X=A⁻¹B.
A⁻¹ = adj A / |A| if |A|≠0. Unique. Properties: A A⁻¹=I, (A⁻¹)⁻¹=A, (AB)⁻¹=B⁻¹A⁻¹, (A^T)⁻¹=(A⁻¹)^T, |A⁻¹|=1/|A|. If |A|=0 no inverse. Solve AX=B ⇒ X=A⁻¹B.
• Formula: A⁻¹ = (1/|A|) adj A if |A|≠0
• Example [[1,2],[3,4]] det=-2 adj=[[4,-2],[-3,1]] A⁻¹=[[-2,1],[1.5,-0.5]]
• Check: A A⁻¹=I
• Properties: (AB)⁻¹=B⁻¹A⁻¹ reverse, |A⁻¹|=1/|A|
• If |A|=0 singular no inverse
• Use: Solve AX=B → X=A⁻¹B
7. Applications - Area of Triangle and System of Equations
Area of triangle with vertices (x1,y1),(x2,y2),(x3,y3) = ½|x1 y1 1 / x2 y2 1 / x3 y3 1| absolute value. If three points collinear, area=0 so det=0 gives collinearity condition. System of linear equations: AX=B, if |A|≠0 unique solution X=A⁻¹B. If |A|=0 check (adj A)B =0? If =0 infinite solutions, if ≠0 no solution. Consistent if solution exists. Cramer's rule is special case for 2×2.
Area = ½|det[[x1 y1 1][x2 y2 1][x3 y3 1]]| absolute. Collinear if det=0. System AX=B: If |A|≠0 unique X=A⁻¹B consistent, If |A|=0 and (adj A)B=0 infinite solutions, if (adj A)B≠0 no solution inconsistent. Check consistency.
• Area triangle = ½| |x1 y1 1 / x2 y2 1 / x3 y3 1| | absolute
• Collinear if area=0 → determinant 0
• System AX=B: |A|≠0 unique X=A⁻¹B
• |A|=0: Check (adj A)B → 0 infinite solutions, ≠0 no solution
• Consistent if solution exists
• Cramer's rule: x=Dx/D, y=Dy/D for 2×2
8. Real Life Usage - Where Determinants Matter
Determinants used in computer graphics to check if transformation flips or scales area, det gives area scaling factor. In engineering, solve circuits, structures. In economics, input-output models. In 3D, volume of parallelepiped = determinant of three vectors. In cryptography, check if matrix invertible for encoding. In machine learning, determinant tells if data invertible. Area calculation in maps, GPS uses determinant for polygon area.
Real: Graphics area scaling factor det, volume of parallelepiped det of vectors, engineering structures, economics input-output, cryptography invertible check, ML covariance det, maps polygon area, GPS.
• Graphics: Det = area scaling, flip if negative
• 3D: Volume parallelepiped = det of 3 vectors
• Engineering: Structures, circuits solve
• Economics: Input-output model
• Crypto: Check invertible matrix for encoding
• Maps: Polygon area via determinant
• ML: Determinant of covariance matrix
• Determinant: |A| number from square matrix, 2×2 ad-bc, |A|≠0 invertible
• Expansion: |A|=Σ a_ij C_ij, choose zero row, Sarrus for 3×3
• Properties: |A^T|=|A|, swap→-det, identical→0, k row→k det, R_i→R_i+kR_j unchanged
• Minor: M_ij delete row i col j det, Cofactor C_ij=(-1)^(i+j)M_ij
• Adjoint: adj A = (Cofactor)^T, A adj A=|A|I
• Inverse: A⁻¹=adj A/|A| if |A|≠0, (AB)⁻¹=B⁻¹A⁻¹, |A⁻¹|=1/|A|
• Area: ½|det[[x1 y1 1][x2 y2 1][x3 y3 1]]|, collinear if 0
• System: AX=B, |A|≠0 unique X=A⁻¹B, |A|=0 check (adj A)B
• Real: Graphics scaling, volume det, crypto invertible, maps area, engineering