Merced College; Don Power

 

LINEAR ALGEBRA - CHAPTER 2, LECTURE

 

2.1  Determinants by Cofactor Expansion; Cramer's Rule

      Hwk:  1, 3ad, 4, 7, 16,  21, 22, 25, 29, T3

 

We will not define a determinant formally until section 4.  In the meantime, we'll use working definitions that are equivalent to the formal definition.

 

a

b

 

 

c

d

 

 
 
 


Def (for 2x2): If A =                             then det(A) = ad – bc

 

Notation:  vertical bars around matrix instead of brackets

 

Minor of aij, called Mij, is the determinant that results from eliminating row i and col j

 

Cofactor if aij, called Cij, is ±Mij, where the sign can be calculated as (–1)i+j

      It is easiest to see the sign by picturing a checkerboard pattern of signs (starting with + in upper left)

 

Cofactor expansion:  det(A) = sum of aijCij along any row or column (i.e.) sum of ± aijMij

      Ex, along row 1:  a11M11 – a12M12 + a13M13 – ...

      Ex, down col 2:  –a12M12 + a22M22 – a32M32 + ...

 

Easiest calculation:  Pick a row or column with the greatest number of 0's  --  this will reduce the number of determinants you have to calculate.

 

Adjoint of a matrix:  adj(A) is the transpose of a matrix whose entries are the cofactors of A
      (the "matrix of cofactors"), or in symbols, [Cij]T

 

Determinant formula for an inverse:  A–1 = 1 / det(A) * adj(A)

      Verify special formula for inverse of a 2x2.

 

Thm 3:  If A is triangular [including diagonal], then det(A) = sum of entries on main diagonal

 

Cramer's Rule for solving a system of equations

      I recommend it for a small system where the coefficients are symbols (rather than numbers).

 

      Ex:  use Cramer's rule to solve   for b and m
      (equations for a least squares line y = mx + b)

 

 

2.2  Evaluating Determinants by Row Reduction         

      Hwk  1a, 2, 3, 12, 14, 16 for #6, 14, 17 for #8, 18, 19

 

Thm:     a.  If A has a row or column of 0's, det(A)=0

                  Proof:  cofactor expansion along row of 0's

                              by signed elementary products:  every elementary product has a factor of 0

                 

            b.  det(A)=det(AT)            

                  Proof:  cofactor expansion of A across 1st row = expansion of AT along 1st column

                              by signed elementary products:  the signed elem. products are the same;

 

Thm:  Determinant of a triangular matrix = product of entries on main diagonal

      Proof:  Repeated cofactor expansions across row 1

                  by signed elementary products:  Every other elementary product must contain at least one 0

     

Thm:  effect of elementary row operations

      Row interchange:  reverse sign       

            Proof by signed elementary products:  reverse 2 factors in every elementary prod

      Multiply a single row or column by a scalar k:  multiply entire determinant by k

            Proof by signed elementary products:  multiply one entry in each elementary prod by k

      Add a multiple of one row to another:  no change          Pr:  omitted

 

Thm:  Same results for multiplication on left by corresponding elementary matrices

      [and, for that matter, for the elementary matrices themselves]

 

Thm:  proportional rows or cols:  det(A) = 0

Pr:  Adding an appropriate multiple of the smaller row (or column) to the larger results in an all-zero row (or column)

 

Technique:  Combination row (or column) reduction with cofactor expansion.

      Aim:  Make all but one entry in a row (or column) 0, then reduce dimension by cofactor expansion

 

2.3  Properties of the Determinant Function
      Hwk  1b, 2, 3, 4bd, 5abe, 6, 9, 12, 13, 14c, 15 (for 14c), 16, 18

 

1.  det(kA) = kn det(A)       Proof:  all n rows are multiplied by k

2.  det(A+B) ¹ det(A) + det(B)            Counterexample in Ex 1;  or let A = B = I2

3.  Thm 1:  but if A, B, C differ only in a single row, and that row of C is sum of rows in A and B,
      then det(A+B) = det(C)            (Also true for columns)

4.  Thm 2:  det(AB) = det(A)det(B)      BIGGIE     Proof requires several steps

      a.  Lemma:  det(EB) = det(E)det(B)            for an elementary matrix E

            Proof:  Thm 2.2.4   (multiplication by elementary matrix is same as elementary row operation)

            Consequence:  also valid for multiplication on the left by multiple elementary matrices

      b.  A is invertible iff ("if and only if") det(A)¹0         BIGGIE

            Pr:  Background:  Let R = rref of A.  Then R=Er...E2E1A

                        Since det(E)¹0, determinants of A and R are either both 0 or both nonzero.

                  (1)  Assume A is invertible.  R must = I, so det (R) and det (A) are nonzero.

                  (2)  Assume det(A)¹0.  Then det(R)¹0, so R cannot have a row of 0's.

                              (contrapositive of row of 0's implies det(A)=0)

                        Therefore, R = I  (Thm 1.4.3:  the rref of an n´n matrix either has a row of 0's or is In)

                        So A is invertible.

      c.   Consequence:  A square matrix with two proportional rows/columns is not invertible

      d.  Proof that det(AB) = det(A)det(B):

(1)  Case 1:  A is not invertible.  Then AB is not invertible (Th1.6.5 AB invertible implies A & B are invertible.  Thus both det(AB) and det(A) equal 0

(2)  Case 2:  A is invertible.  Write A as product of elementary matrices, multiply by B on right, take determinant.

 

5.  det(A-1) = 1 / det(A)     Pr:  take determinent of A-1A = I

6.  Systems of form Ax = λx           Really homogeneous

      (λI-A)x=0

      det(λI-A)=0    iff system has nontrivial solutions      "Characteristic equation" of A

            λ = eigenvalue;  the corresponding solutions are "eigenvectors"

7.  More equivalent statements in "big theorem:"  add det(A)¹0 as equiv to A is invertible

 

 

2.4  The Determinant Function:  Combinatorial Approach 

      Hwk  1ab, 2ab, 5, 9, 13b, 17, 19, 22, 23a

 

Def:  Permutation of the set of integers (1, 2, ..., n} is an arrangement, without omissions or repetitions

Ex:  List some permutations of {1, 2, 3, 4, 5}

Counting note:  number of permutations is n! (why?)

 

Def:  an inversion occurs in a permutation whenever a larger integer precedes a smaller one

Ex:  identify inversions in the prev example

How to count inversions:  For each integer, count smaller integers to the right; add the results

      Ex:  prev

 

Def:  A permutation is even or odd according to whether the number of inversions is even or odd

      Ex: prev

 

Def:  An elementary product from an n´n matrix A is a product of n factors from A, no two of which come from the same row or the same column

      Arrange the factors of an elementary product in order by rows, and

Consider the permutation of the column numbers.

A signed elementary product is positive or negative according to whether the permutation of the columns is even (+) or odd (-)

 

Def:  Let A be a square matrix.  The determinant of A, det(A) = sum of the signed elementary products of A

      The determinant is the single number result

Vertical bracket notation for the unevaluated determinant

Calculation of 2´2 det

      Ex

Calculation of 3´3

      Ex, with 6-arrow shortcut. 

Caution:  applies only to 3´3 (a 4´4 would need 4! = 24 diagonals, but only 8 exist)

How to write symbolically with summation symbol

      Text

 

 

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