Determinants

1
Chapter 6.7

• Determinants

2

• In this chapter all matrices are square for
  example 1x1 (what is a 1x1 matrix, in fact?),
  2x2, 3x3
• Our goal is to introduce a new concept, the
  determinant, which is only defined for square
  matrices, yet any size square matrices

3

• The determinant is, first of all, just a number
  and, since we want to have a natural definition
  for it, we say that for 1x1 matrices the
  determinant is EXACTLY that number

4

• Examples
• Notation we use either the keyword det in
  front of the matrix OR we replace the brackets
  (or parenthesis, if applicable) with vertical bars

5

• Careful for 1x1 matrices DO NOT confuse the
  vertical bars notation with the absolute value
  notation (the context will tell you whether the
  number is assumed to be a matrix, in which case
  its determinant, or a number, in which case its
  absolute value)

6

• What about bigger matrices? The idea is to define
  a bigger matrix determinant based on smaller
  (sub-)matrices determinant
• Take a 2x2 matrix a smaller matrix is a 1x1
  matrix, whose determinant we know right now
  hence we can define the 2x2 matrix determinant

7

• Choose a row (usually the first one)
• take the first entry in the row, and remove from
  the matrix the column corresponding to it and the
  chosen row whats left is a matrix of order 2-11

8

• take the determinant of this matrix (we know how
  to compute it!) we call this the minor and its
  usually denoted by capital letter with the
  initial entry indices (in our case 11)

9

• The last thing we do for the 11 index is to build
  the cofactor, which is the minor multiplied by
  (-1) to the power (sum of indices), in our case
  112 the usual notation for cofactor is c with
  original indices

10

• Take now the second entry and build its cofactor
  remove the second column and the first row
  whats left is a 1x1 matrix, namely
• compute the minor

11

• Get the cofactor
• we exhausted the row, and now we construct the
  sum

12

• For the 2x2 matrix we get, in fact, an easy to
  remember formula product of the first diagonal
  (NW-SE) minus product of the second diagonal
  (NE-SW)
• Example

13

• Things get a bit more complicated as we go to 3x3
  matrices we use the VERY SAME TECHNIQUE, though
• take the matrix

14

• Choose a row - again, usually we take the first
  row, so lets go with that one
• take first entry and compute its cofactor remove
  the first column and the first row, and we get
  the following matrix

15

• We compute this smaller (2x2) matrix determinant
  (because we know now how!) and so we get the
  minor of index 11
• finally, the cofactor

16

• Next entrys cofactor (less talk, just
  computation)

17

• So, the cofactor is
• and, for the last entry of this row, voila the
  cofactor

18

• Hence the determinant for our 3x3 matrix is

19

• There actually is a way of remembering even this
  formula, reminiscent of the 2x2 matrix
  determinant again, we have a first diagonal
  (NW-SE), but also 2 first half-diagonals the
  product of entries on each gets added we have a
  second diagonal (NE-SW) and 2 second
  half-diagonals, and the product of entries on
  each, respectively, gets substracted (for
  alternate description please read at the bottom
  of page 280)

20

• Example

21

• One interesting issue as mentioned, you could
  choose any row you want (and, in fact, if you
  look sidewise, you could do a very similar
  thing for columns!) but the process is pretty
  complex, right? So how can we be so sure we get
  the same number all the time? Well, be assured -
  it really works, regardless of row (or of column,
  in fact)

22

• Why would you choose a different row? For
  example, you have a lot of 0 (zero) entries in
  that row since you got to multiply those entries
  with the respective cofactors, 0 times anything
  is 0! So we dont need to compute those cofactors!

23

• Example
• if you choose the first row, you need to compute
  all three cofactors but if you choose the second
  row, you only need compute the second entrys
  cofactor!

24

• the determinant is hence (Im mentioning the
  other 2 cofactors, but, as you see, they get
  multiplied by 0, so we dont care what values
  they have)

25

• As an exercise, try to prove that, by choosing
  the second row in a 2x2 matrix you get the same
  number (you can work with an actual matrix, or
  try the general form, with as)

26

• Complicated as it was for 3x3 matrices, you can
  see now how complicated it could get even further
  (4x4, 5x5 and so on) still the method still
  works, and the moment you know how to compute a
  3x3 matrix determinant, you can compute the 4x4
  matrix one the moment you know how to compute a
  4x4 matrix determinant, you can compute the 5x5
  matrix one

27

• But computing a determinant is not always a
  hideously long and intricate task, because the
  way we compute this number leaves a few backdoors
  open
• For instance
• 1. If each of the entries in a certain row (or
  column) of the matrix is 0, then its determinant
  is 0 (remember the example above? What if the 4
  was also a 0?)

28

• 2. If two rows (or columns) are identical, then
  the determinant is 0
• 3. If the matrix is upper/lower triangular (in
  particular, if it is diagonal) then the
  determinant is equal to the product of the main
  (first) diagonal entries

29

• 4. If one modifies the matrix by adding a
  multiple of one row to another row (same for
  columns), the determinant doesnt change - here
  you see the connection to elementary matrix
  operations!

30

• 5. If one interchanges two rows (or columns) the
  determinant changes sign
• 6. If one multiplies the entries of a row (or
  column) by a number, the determinant gets
  multiplied by that number

31

• 7. If one multiplies the matrix by a scalar
  (which, if you remember, means multiplying all
  elements by that number - all rows, that is, and
  see 6.) then the determinant gets multiplied by
  that number as many times as many rows it has
  (its size for a 2x2 matrix, twice 3x3 matrix,
  thrice and so on)

32

• 8. If one multiplies two matrices, the
  determinant of the product is the product of the
  determinants
• 9. The determinant of the transpose of a mtrix
  equal the determinant of the original matrix

33

• One last thing as you can see, its very
  convenient to have an upper triangular matrix (or
  lower, or diagonal) on the other hand, when
  reducing a matrix (a square matrix, that is)
  thats exactly what we get

34

• So why not use this? As you can see, one
  elementary operation doesnt change the
  determinant (adding a multiple of a row to
  another one probably the most important one see
  4.) a second one only changes its sign
  (interchanging two rows see 5.) the last one
  multiplies the determinant by a controllable
  quantity (multiplying by that quantity a certain
  row see 6.)

35

• Heres an example
• think of this as factoring out the 3 out of the
  second row

36

• (we now have an upper triangular matrix, so we
  can stop here - or, if you can waste time, NOT
  DURING THE EXAM! you can continue reducing the
  matrix by factoring out the -7 out of the
  second row etc)
• This method is called triangulation (for obvious
  reasons!)

37

• Its especially useful for higher order matrices
  (4x4, 5x5, etc) since we dont have to compute
  many complex cofactors, but rather use simple
  elementary operations both methods take time,
  though (in general, expect to waste a lot of time
  when computing a large matrix, with few 0 )