Determinants» a b : det c d = a c b d = ad bc a b c : det 4d e f5 = g h i a b c d e f g h i = +aei + bfg + cdh ceg afh bdi M100 Vector Geometry and Linear Algebra 1
a b c d 4 4 : det e f g h 4 i j k l 5 = m n o p a b c d e f g h i j k l m n o p = +afkp+agln+ahjo+belo+bgip+bhkm+cejp+cflm+chin+dekn+dfio+dgjm -aflo-agjp-ahkn-bekp-bglm-bhio-celn-cfip-chjm-dejo-dfkm-dgin M100 Vector Geometry and Linear Algebra
5 5 : a b c d e f g h i j k l m n o p q r s t u v w x y = +agmsy+agntw+agorx+ahltx+ahnqy+ahosv+ailry+aimtv+aioqw+ajlsw+ajmqx+ajnrv -agmtx-agnry-agosw-ahlsy-ahntv-ahoqx-ailtw-aimqy-aiorv-ajlrx-ajmsv-ajnqw +bfmtx+bfnry+bfosw+bhksy+bhntu+bhopx+biktw+bimpy+bioru+bjkrx+bjmsu+bjnpw -bfmsy-bfntw-bforx-bhktx-bhnpy-bhosu-bikry-bimtu-biopw-bjksw-bjmpx-bjnru +cflsy+cfntv+cfoqx+cgktx+cgnpy+cgosu+cikqy+ciltu+ciopv+cjksv+cjlpx+cjnqu -cfltx-cfnqy-cfosv-cgksy-cgntu-cgopx-ciktv-cilpy-cioqu-cjkqx-cjlsu-cjnpv +dfltw+dfmqy+dforv+dgkry+dgmtu+dgopw+dhktv+dhlpy+dhoqu+djkqw+djlru+djmpv -dflry-dfmtv-dfoqw-dgktw-dgmpy-dgoru-dhkqy-dhltu-dhopv-djkrv-djlpw-djmqu +eflrx+efmsv+efnqw+egksw+egmpx+egnru+ehkqx+ehlsu+ehnpv+eikrv+eilpw+eimqu -eflsw-efmqx-efnrv-egkrx-egmsu-egnpw-ehksv-ehlpx-ehnqu-eikqw-eilru-eimpv
+ a 4 c b d 5 ad bc + + a d 4 g + h c a f d 5 i g b b e e h + aei + bfg + cdh ceg afh bdi M100 Vector Geometry and Linear Algebra 4
+ + + + a b c d a b c e f g h e f g i j k l i j k 4 5 m n o p m n o?? NO! Only gives eight terms! M100 Vector Geometry and Linear Algebra 5
Cofactors a b c d e f g h i = +aei + bfg + cdh ceg afh bdi = a(ei fh) + b(fg di) + c(dh eg) = a(cofactor of a) + b(cofactor of b) + c(cofactor of c) = a e h f i + b f i d g + c d g e h = a e h f i + b d g f «i + c d g e h M100 Vector Geometry and Linear Algebra
a b c d e f g h i j k l m n o p = +afkp+agln+ahjo+belo+bgip+bhkm+cejp+cflm+chin+dekn+dfio+dgjm -aflo-agjp-ahkn-bekp-bglm-bhio-celn-cfip-chjm-dejo-dfkm-dgin =a(fkp + gln + hjo flo gjp hkn) + (terms without a) f g h So the cofactor of a is fkp + gln + hjo flo gjp hkn = j k l n o p e g h The cofactor of b is elo + gip + hkm ekp glm hio = i k l m o p M100 Vector Geometry and Linear Algebra
Definition If row i and column j of an n n matrix A are deleted, the remaining entries collapse to form an (n 1) (n 1) matrix The determinant of this submatrix is called the ij-th minor of A, denoted M ij If the minor M ij is multiplied by ( 1) i+j (this is +1 if i and j are both even or both odd, and is 1 if one of i and j is even and the other odd), then we obtain the i, j-cofactor, C ij = ( 1) i+j M ij Then the determinant of A is defined to be the cofactor expansion along the first row: det A = a 11 C 11 + a 1 C 1 + + a 1n C 1n M100 Vector Geometry and Linear Algebra 8
5 1 Example: If A = 40 5, then the minors of the first row are 4 0 M 11 = 0 =, M 1 = 0 4 = 8, M 1 = 0 4 0 = 1 And then the cofactors are C 11 = ( 1) M 11 = +, C 1 = ( 1) M 1 = 8, C 1 = ( 1) 4 M 1 = 1 Finally, the determinant of A is det(a) = a 11 C 11 + a 1 C 1 + a 1 C 1 = ()() + ( 5)(8) + (1)( 1) = 1 40 1 = 40 M100 Vector Geometry and Linear Algebra 9
Theorem The determinant can also be calculated by expansion along the i-th row: or the j-th column: det A = a i1 C i1 + a i C i + + a in C in det A = a 1j C 1j + a j C j + + a nj C nj Example: Again let A = 5 1 40 5, but expand along the second row: 4 0 det(a) = a 1 C 1 + a C + a C = (0)( 1) 5 1 0 + ()(+1) 1 4 + ( )( 1) 5 4 0 = 0 + (4 4) + (0 + 0) = 40 M100 Vector Geometry and Linear Algebra 10
The pattern of + and signs given by ( 1) i+j produces this checkerboard pattern: 4 + + +? + + + + + + + + + +? + 5 Note that the diagonal entries are all + M100 Vector Geometry and Linear Algebra 11
Entries and cofactors from different rows Suppose we take entries from one row of a matrix, and cofactors from a different row: a i1 C j1 + a i C j + a i C j + + a in C jn Then the result is always 0 For example, if A = a b c 4d e f5, then g h i a 11 C 1 + a 1 C + a 1 C = a b h c i + b a g c i c a g b h = a(bi ch) + b(ai cg) c(ah bg) = abi + ach + bai bcg cah + cbg = 0 M100 Vector Geometry and Linear Algebra 1
So we have: Theorem For any n n matrix A, a i1 C j1 + a i C j + a i C j + a in C jn = ( det A if i = j, 0 if i j We can interpret this as a certain matrix product M100 Vector Geometry and Linear Algebra 1
Cofactor matrix and adjoint matrix If we calculate all the cofactors of A and put them into a matrix, we get the cofactor matrix: C 11 C 1 C 1 C 1n C 1 C C C n cof(a) = C 1 C C C n 4 5 C n1 C n C n C nn and the transpose of the cofactor matrix is the adjoint matrix: C 11 C 1 C 1 C n1 C 1 C C C n adj(a) = (cof(a)) T = C 1 C C C n 4 5 C 1n C n C n C nn M100 Vector Geometry and Linear Algebra 14
Let B = adj(a) = cof(a) T, so that b ij = C ji Then ij-th entry of AB = a i1 b 1j + a i b j + a i b j + + a in b nj = a i1 C j1 + a i C j + a i C j + + a in C jn ( ) det A if i = j, = = ij-th entry of (det A) I n 0 if i j M100 Vector Geometry and Linear Algebra 15
The adjoint formula for A 1 Theorem Let A be an n n matrix 1 A adj(a) = adj(a) A = (det A)I n A is invertible if and only if det A 0 If det A 0, then A 1 = 1 det A adj(a) M100 Vector Geometry and Linear Algebra 1
Example 1 0 1 Let A = 40 15 Then the cofactors are: 0 1 C 1,1 = + 1 1 = C 1, = 0 1 0 = 0 C 1, = + 0 0 1 = 0 C,1 = 0 1 1 = 1 C, = + 1 1 0 = C, = 1 0 0 1 = 1 C,1 = + 0 1 1 = C, = 1 1 0 1 = 1 C, = + 1 0 0 = So the cofactor matrix is cof(a) = C 1,1 C 1, C 1, 4C,1 C, C, 5 = C,1 C, C, 0 0 4 1 15 1 M100 Vector Geometry and Linear Algebra 1
And the adjoint is the transpose: adj(a) = (cof(a)) T = 1 40 15 0 1 Multiplying A by its adjoint gives: A adj(a) = 1 0 1 1 40 1540 15 = 0 1 0 1 0 0 40 05 = I 0 0 The determinant of A is det A =, so A 1 = 1 1 1/ / adj(a) = 40 / 1/5 0 1/ / Exercise: Check this by the [A I] method M100 Vector Geometry and Linear Algebra 18