Math 19b: Linear Algebra with Probability Oliver Knill, Spring 2011Lecture 24: DeterminantsIn this lecture, we define the determinant for a general n × n matrix and look at the Laplaceexpansion method to compute them. A determinant attaches a number to a square matrix, whichdetermines a lo t about the matrix, like whether the matrix is invertible.The 2 × 2 caseThe determinant o f a 2 × 2 matr ixA ="a bc d#is defined as det(A) = ad − bc.We have seen that this is useful for the inverse:A−1="d −b−c a#ad − bc.This formula shows:A 2×2 matrix A is invertible if and only if det(A) 6= 0. The determinant determinesthe invertibility of A.1 det("5 42 1#) = 5 · 1 − 4 · 2 = −3.We also see already that the determinant changes sign if we flip rows, t hat the determinant islinear in each of the rows.We can write the formula as a sum over all permutations of 1, 2. The first permutation π = (1, 2)gives the sum A1,π(1)A2,pi(2)= A1,1A2,2= ad and the second permutation π = (2, 1) g ives thesum A1,π(1)A2,π(2)= A1,2− A2,1= bc. The second permutation has |π| upcrossing and the sign(−1)|π|= −1. We can write the above formula asPπ(−1)|π|A1π(1)A2π(2)."a bcd#−"abc d#The 3 × 3 caseThe determinant o f a 3 × 3 matrixA =a b cd e fg h iis defined as aei + bf g + cdh − ceg − bdi − af h.We can write this as a sum over all permutations of {1, 2, 3}. Each permutation produces a”pattern” along we multiply the ma t rix entries. The patterns π with an even number |π| ofupcrossings are taken with a positive sign the other with a negative sign.a b cde fg h i+ab cd e fg h i+a bcd e fg h i−a bcd e fg h i−ab cd e fg h i−a b cd e fg h i2 det(2 0 41 1 11 0 1) = 2 − 4 = −2.The general definit i onA permutation is an invertible transformation {1, 2, 3, . . . , n} onto itself. We canvisualize it using the permutation matrix which is everywhere 0 except Ai,π(i)= 1.There are n! = n(n − 1)(n − 2) · · · 2 · 1 p ermutations.3 For π = (6, 4, 2, 5, 3, 1) if π(1) = 6, π(2) = 4, π(3) = 2, π(4) = 5, π(5) = 3, π(6) = 1 we havethe permutation matrixPπ=111111.It has |π| = 5 + 2 + 3 + 1 + 1 = 12 up-crossings. The determinant of a matrix which haseverywhere zeros except Aiπ(j)= 1 is the number (−1)|πwhich is called the sign of thepermutation.The determinant of a n × n matrix A is defined as the sumXπ(−1)|π|A1π(1)A2π(2)· · · Anπ(n),where π is a permutation of {1, 2, . . . , n } and |π| is the number of up- cro ssings.4det(A) = det(0 0 0 0 020 0 0 3 0 005 0 0 0 00 0 0 07 00 0 11 0 0 013 0 0 0 0) = 2 ∗ 3 ∗ 5 ∗ 7 ∗ 11 ∗ 13 .The determinant of an upper triangular matrix or lower triangular matrix is theproduct of the diagonal entries.Laplace expansionThis Laplace expansion is a convenient way to sum over all permutations. We group the permu-tations by taking first all the ones where the first entry is 1, then the one where the first entry is 2etc. In that case we have a permutation of (n − 1) elements. the sum over these entries producesa determinant of a smaller matrix.For each entry aj1in the first column form the (n − 1) × (n −1) matrix Bj1which does not containthe first and j’th row. The determinant of Bj1is called a minor.Laplace expansion det(A) = (−1)1+1A11det(Bi1) + · · · + (−1)1+nAn1det(Bn1)5 Find the determinant of0 0 7 0 0 08 0 0 0 0 00 0 0 0 0 13 0 0 1 0 00 0 0 0 5 00 2 0 0 0 0.We have two nonzero entries in the first column.det(A) = (−1)2+18det0 7 0 0 00 0 0 0 10 0 1 0 00 0 0 5 02 0 0 0 0+ (−1)4+13det0 7 0 0 00 0 0 0 00 0 0 0 10 0 0 5 02 0 0 0 0= −8(2 ∗ 7 ∗ 1 ∗ 5 ∗ 1) + −3(2 ∗ 7 ∗ 0 ∗ 5 ∗ 1) = −5606 Find the determinant of0 0 0 1 0 0 00 0 0 0 0 1 00 1 0 0 0 0 00 0 1 0 0 0 00 0 0 0 0 0 11 0 0 0 0 0 00 0 0 0 1 0 0.The answer is −1.7 Find the determinant ofA =3 2 3 0 0 00 4 2 0 0 00 0 5 0 0 00 0 0 0 1 00 0 0 0 0 10 0 0 1 0 0.Answer: 60.Homework due April 6, 20111 Find the determinant of t he following matrix0 3 0 0 0 00 0 4 0 0 01 0 3 4 7 12 0 0 1 0 00 0 0 4 1 00 0 6 0 0 0.2 Give the reason in terms o f permutations why the determinant of a partitioned matrix"A 00 B#is the product det(A)det(B).Example det(3 4 0 01 2 0 00 0 4 −20 0 2 2) = 2 · 12 = 24.3 Find the determinant of t he diamond matrix:A =0 0 0 0 8 0 0 0 00 0 0 8 8 8 0 0 00 0 8 8 8 8 8 0 00 8 8 8 8 8 8 8 08 8 8 8 8 8 8 8 80 8 8 8 8 8 8 8 00 0 8 8 8 8 8 0 00 0 0 8 8 8 0 0 00 0 0 0 8 0 0 0 0.Hint. Do not compute too much. Investigate what happens with the determinant if youswitch two