Determinants

Everything You Ever Wanted to Know (and probably more…)

 

Basics of Determinants  in a nutshell

 

determinants only apply to square (nxn) matrices

the result is a number. 

whether the number is 0 or not tells you a lot about the matrix and any

system it is the coefficient matrix of

 

Computing them

            2 methods

                        criss-cross

                        expansion about a row

 

            (criss cross only works on 2x2 and 3x3)

 

            there are n! terms in the expansion of an nxn matrix  (3!=3*2*1 etc)

            easy in Maple, Matlab, many calculators

 

Mathematical Properties

 

     4 "Basic Properties"

            Det( In) = 1    (identity matrix)

            you can factor a constant out of a single row

            you can swap any two rows if you put a – sign in front of the determinant

            you can add a multiple of one row to another and the determinant does not 

change

 

     What does this mean for practical purposes?

 

            if you have a row of 0s, the determinant is 0

            if you have two identical rows the determinant is 0

            if you have a diagonal matrix, the determinant is the product of the

                        diagonal entries

if you have an upper (or lower) triangular matrix, the determinant is the  

            product of the diagonal entries

the determinant of a matrix is a constant times the determinant of

            its Final Form

moral: many determinants can be done just by looking at them!

 

Implications For Linear Systems  Ax = b (n eqns, n unknowns )

           

            Final Form of A has row of 0s ó Det(A) = 0

            rank of A < n  ó Det(A) = 0

            rank of A = n  ó Det(A) ≠ 0

            A-1 exists ó Det(A) ≠ 0

            A-1 doesn't exist  ó Det(A) = 0

            Ax = 0 has nontrivial solutions ó Det(A) = 0

 

Example 1

            For a 5x5 matrix A,it has been computed that Det(A) = -4.  What information

            can be deduced from this?

 

            ans:  

                        rank(A) = 5

                        A-1     exists

                        Final Form of A  is  I5

                        Ax = b has a unique solution  

                        Ax = 0 has only the trivial solution

 

Example 2

 

            A  6x6 matrix A has determinant 0.  What can be deduced from this?

 

            ans:

                        rank of A is 5 or less

                        A-1     does not exist

                        Ax = b may or may not have solutions

                        Ax = 0  has nontrivial solutions

                        the Final Form of A at least one row of 0s in it

 

Example 3

            A 3x3 matrix A has determinant 0. What geometric information can be deduced from this?

 

            ans:  all three vectors that comprise the rows lie in the same plane

 

Notes: also used for Cross Products, esp in Physics courses

 

            various notations:   Det(A) = D(A) = |A|

 

More Advanced Tidbits

 

            Det(A)   =  Det(At)   (t  means transpose)

 

            So???  You ask….  this means everything you know about rows applies to columns!!