Determinant of an n n matrix

Transcription

1 Determinant of an n n matrix The determinant is a function from the set of n n matrices to R. (Or from n-tuples of row vectors in R n, to R). For us the determinant is defined by the first row expansion, explained later. The definition is recursive. Notation: det A or A. A = [ ] det [ ] 2 5 = Definition For a 1 1 matrix [a] a = a

2 3 3 determinant, first row expansion The determinant of a 3 3 matrix is defined by the first row expansion. This is an sign-alternating sum of first row entries times 2 2 determinants. First row entries are in blue. Each blue entry is multiplied by a 2 2, obtained by crossing out the row and column of the blue entry. a b c A = g h i these matrices are just pictures g h i g h i g h i g h i = a e f h i b d f g i + c d e g h This defines a 3 3 determinant, using 2 2 determinants.

3 2 2 determinant, first row expansion Blue gives first row entries. Signs alternate. To get the 1 1 determinant to multiply by a first row entry, cross out the row and the column of the entry. [ ] [ ] a b a b just pictures c d c d a b c d = a d b c = ad bc.

4 3 3 determinant, first row expansion, continued Now that we know 2 2 determinants, we finish the previous 3 3: g h i = a e f h i b d f g i + c d e g h = a(ei fh) b(di fg) + c(dh eg) = aei afh bdi + bfg + cdh ceg.

5 3 3 determinant, second column expansion Can expand along any row or column and the answer is the same. a b c A = g h i a b c a b c a b c just pictures g h i g h i g h i g h i = b d f g i + e a c g i h a c d f Answer matches first row expansion. = b(di fg) + e(ai cg) h(af cd) = bdi + bfg + eai ecg haf + hcd

6 General sign pattern The assign associated with the (i, j)-th entry is ( 1) i+j.

7 Scaling a row scales the determinant Why it works: multiply first row by m and expand along first row: ma mb mc g h i = ma e f h i mb d f g i + mc d e g h ( = m a e f ) h i b d f g i + c d e g h = m g h i Determinant has been multiplied by m.

8 Swapping rows changes determinant by a sign: Alternating property Why it works for 2 2: check directly c d a b = cb ad = (ad bc) = a c b d Why it works for 3 3: swap rows 2 and 3. Use just-verified alternating property for 2 2: g h i = a h i e f b g i d f + c g h d e ( = a e f ) h i b d f g i + c d e g h = g h i

9 The determinant of a matrix with two equal rows, is zero If we swap the two identical rows, the determinant changes by a sign using the alternating property. However since the swapped rows are identical, the determinant did not change because the matrix did not change. The only number that doesn t change when you negate it, is zero. Hence the determinant is zero.

10 Determinant is additive in any given row First row varies; other rows fixed. g h i + a b c g h i = a e f h i b d f g i + c d e g h + a e f h i b d f g i + c d e g h = (a + a ) e f h i (b + b ) d f g i + c + c ) d e g h = a + a b + b c + c g h i

11 Determinant is linear in its first row Let n = 3 and consider the determinant as a function of three row vectors u = [a, b, c], v = [d, e, f ], w = [g, h, i]. Let T : R 3 R be the function of a single vector u R 3 defined by u T (u) = det v with v, w fixed. w Claim: T is a linear function. Check additive property. Let u = [a, b, c ]. Then u + u = [a, b, c] + [a, b, c ] = [a + a, b + b, c + c ]. We have u + u T (u + u a + a b + b c + c ) = det v = w g h i a b c u = g h i + g h i = det v w + det u v w = T (u) + T (u )

12 Determinant is linear in its first row, continued n = 3: row vectors u = [a, b, c], v = [d, e, f ], w = [g, h, i]. Recall T : R 3 R is the function of a single vector u R 3 defined by u T (u) = det v with v, w fixed. w Check T has the scalar multiplication property. mu = m[a, b, c] = [ma, mb, mc]. mu ma mb mc T (mu) = det v w = g h i = m = mt (u). g h i Therefore this function T is linear. Determinant is linear in its first row. This works for any row. Determinant is multilinear.

13 (i) Scaling a row of a matrix, scales its determinant. (ii) Viewing the determinant as a function of a sequence of n row vectors in R n, if we fix all of those vectors but one, the resulting function of a single vector, is additive. Properties (i) and (ii) together say that the determinant is a multilinear function. (iii) Exchanging two rows of a matrix, changes its determinant by a sign (negation). The determinant is an alternating function (of n row vectors in R n ). Theorem The determinant is the unique multilinear alternating function of n row vectors in R n to R, which sends I n to 1.

14 Adding a multiple of one row to another, doesn t change determinant Proof for 3 3: Let u, v, w be rows of a 3 3 matrix. Do the row operation R 3 R 3 + cr 1. Get the matrix with rows u, v, w + cu. Take the determinant: u u u det v = det v + det v w + cu w cu u u = det v + c det v w u u u = det v + c 0 = det v w w because det is additive in the 3rd row, det scales in the 3rd row, and if two rows are equal then the determinant is zero.