MATH 1046 Determinants (Section 4.2) Alex Karassev

Determinant in the 2x2 case Recall: A is invertible if and only if det A ≠ 0

Determinant in the 2x2 case Each product contains exactly one element from every column and every row

Determinants in the 3x3 case Products of elements of the matrix Every product contains exactly one element from each row and each column Signs of products must alternate according to a special rule

Entries from the first row: a11 a12 a13 a11a22a33 a12a23a31 a13a22a31 a11 a21 a11 a11a23a32 a12a21a33 a13a21a32

The first indices in all products are always 1,2,3 while the second indices form all possible rearrangements of 1,2,3 a11a22a33 a12a23a31 a13a22a31 a11 a21 a11 a11a23a32 a12a21a33 a13a21a32

How do we choose signs? a11a22a33 a12a23a31 a13a22a31 a11 a21 a11

Permutations A permutation i1 i2 … in of 1,2,…,n is a rearrangement of 1,2,…,n There are 12 … (n-1) n = n! permutations We can assign a signature to each permutation as follows: sgn(i1 i2 … in ) = (-1)k, where k is the number of two-element swaps (called transpositions) required to get i1 i2 … in from 1 2 … n

Remarks Choice of k is not unique For example: 4 2 1 3 can be obtained from 1 2 3 4 in many different ways:

Remarks Choice of k is not unique For example: 4 2 1 3 can be obtained from 1 2 3 4 in many different ways: 1 2 3 4

Remarks Choice of k is not unique For example: 4 2 1 3 can be obtained from 1 2 3 4 in many different ways: 1 2 3 4

Remarks Choice of k is not unique For example: 4 2 1 3 can be obtained from 1 2 3 4 in many different ways: 1 2 3 4 → 3 2 1 4

Remarks Choice of k is not unique For example: 4 2 1 3 can be obtained from 1 2 3 4 in many different ways: 1 2 3 4 → 3 2 1 4

Remarks Choice of k is not unique For example: 4 2 1 3 can be obtained from 1 2 3 4 in many different ways: 1 2 3 4 → 3 2 1 4 → 4 2 1 3 (k=2)

Remarks Choice of k is not unique For example: 4 2 1 3 can be obtained from 1 2 3 4 in many different ways: 1 2 3 4 → 3 2 1 4 → 4 2 1 3 (k=2) 1 2 3 4 → 2 1 3 4 → 2 3 1 4 →4 3 1 2 → 4 2 1 3 (k=4)

Remarks Choice of k is not unique For example: 4 2 1 3 can be obtained from 1 2 3 4 in many different ways: 1 2 3 4 → 3 2 1 4 → 4 2 1 3 (k=2) 1 2 3 4 → 2 1 3 4 → 2 3 1 4 →4 3 1 2 → 4 2 1 3 (k=4) However, in both cases the parity of k is the same, so sgn ( 4 2 1 3) = (-1)2 = (-1)4 = 1 It can be shown in general that signature is well-defined (i.e. independent on k)

Definition of Determinant

Determinant – 3x3 case

Example

Example

Cofactor expansion – 3x3 case

Cofactor expansion – 3x3 case

Minors and cofactors Let Aij be the (n-1)x(n-1) matrix obtained from A by deleting i-th row and j-th column The (i,j)-minor of A is Mij = det Aij The (i,j)-cofactor of A is Cij = (-1)i+j Mij

Minors and cofactors

Laplace cofactor expansion theorem For all i,j = 1,2,…n we have Cofactor expansion along the i-th row: Cofactor expansion along the j-th column: (see text for the proof)

Signs (-1)#row + #column

Example

Example Using cofactor expansion along the 3rd row:

Exercise Make sure that cofactor expansion along the 2nd column gives the same answer

Theorem Determinants of lower or upper triangular matrices are equal to the products of the diagonal elements Consequently, the determinant of a diagonal matrix is the product of its diagonal elements

4x4 case

Proof Follows from cofactor expansions along the first row or the first column and induction on the size of a matrix

Determinants and the elementary row operations Let B be obtained from A using one of the following operations: Swap two rows: Ri  Rj Multiply a row by a nonzero number: cRi Add a multiple of one row to another row: Rj → Rj + cRi

Determinants and the elementary row operations Swap two rows: Ri  Rj det B = - det A Multiply a row by a nonzero number: cRi det B = c det A Add a multiple of one row to another row: Rj → Rj + cRi det B = det A

Corollary Suppose that A ~ B. Then det A = 0 if and only if det B = 0

Proof – swapping two rows B is obtained from A by swapping rows 1 and 2

Proof - multiplying a row by a nonzero scalar B is obtained from A by cR1

Proof – adding a multiple of one row to another row B is obtained from A by R1 → R1 + cR2

Corollary A is invertible if and only if det A ≠ 0 (indeed, A is invertible if and only if the r.r.e.f. of A is I) Example: the following matrix is invertible, since its determinant is 20 ≠ 0

Corollary If A has a row (or column) consisting entirely of zeros or two coinciding rows (or columns) then det A = 0

Example Since the following matrix is not invertible, its determinant is zero:

Determinants of elementary matrices Recall: Ri  Rj: Eij cRi: Ei(c) Rj → Rj + cRi : Eij(c)

Determinants of elementary matrices Since the elementary matrices are obtained from I using elementary row operations and det I = 1, we have the following formulas det(Eij)= -1 det(Ei(c))= c det(Eij(c))= 1

Lemma Let A and B be two n x n matrices. Then A or B is not invertible if and only if AB is not invertible

Proof Consider the following sequence of equivalent statements: AB is not invertible null(AB) ≠ 0 There exists x ≠ 0 in Rn such that (AB)x = 0 So A(Bx) = 0 If y = Bx = 0 then null(B) ≠ 0 so B is not invertible Otherwise, y ≠ 0 and Ay = 0 so null(A) ≠ 0, and hence A is not invertible

Corollary det (AB) = 0  det A =0 or det B =0

Determinants and matrix multiplication Lemma If E is an elementary matrix and A is any n x n matrix then det (EA) = det(E)det(A) Proof: follows from the effect of elementary row operations on determinants and from the formulas for the determinants of elementary matrices

Determinants of products For n x n matrices A and B det (AB)= det(A)∙det (B)

Proof If det A =0 it is the consequence of the previous corollary, so suppose A is invertible Then A is a product of elementary matrices, A = A1 A2 … As, so we have: det (AB) = det (A1 A2 … As B) = (by Lemma) = det (A1) det (A2 … As B) = (applying Lemma again) = det (A1) det (A2) det (A3… AsB) = (and so on) = det (A1) det (A2) … det (As-2) det (As-1) det (As) det B = (applying lemma in the other direction) det (A1)det(A2)…det(As-2) det(As-1As) det B = (and so on) =det(A1 A2 …As) det B = det(A)det(B)

Corollary If A is invertible then det (A-1) = 1/det(A) Proof 1 = det I = det(AA-1) = det(A) det(A-1) Therefore det(A-1) = 1 / det (A)

Example Let I be the 2 x 2 identity matrix Then det (I+2I) = det (3I) = 33 =9, while det(I)=1 and det(2I) = 22 =4 So in general det (A+B) ≠ det(A) + det (B)

Determinants of scalar multiples For an n x n matrix A, det (cA) = cn det (A) Proof cA = (cI)A and therefore det(cA) = det((cI)A) = det(cI) det(A) = cn det(A)

Determinant of transpose det (AT) = det (A)

Proof Note that (AT)ij = (Aji)T Using cofactor expansion along the first row of B=AT and induction on the size of the matrix we get:

Determinants and matrix operations - summary det AB = det A det B = det BA det (cA) = cn det A det A-1 = 1 / det A det AT = det A

Exercise Find the following determinants:

Geometry of determinants Let A be a 2 x 2 matrix |det(A)| is the area of the parallelogram spanned by the columns of A Similar interpretation exists in higher dimensions

Geometry of determinants |det A| Exercise: prove it! b c a

Example Find the area of the triangle with vertices A(1,1), B(2,2),C(0,5) Solution: Since the area of triangle ABC is half the are of the parallelogram spanned by vectors AB and AC, we have:

Determinants and systems of linear equations To solve Ax = b we can use Cramer’s rule: Let Ai(b) be the matrix obtained by replacing the i-th column of A with b, i=1,2,…,n Then the solution of the system is given by xi = det (Ai(b)) / det A, i = 1,2,…,n

Example Solve the following system using Cramer’s rule:

Solution