Chapter 6.7

Determinants

• 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

• 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:

111111 )det( aaa

• 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

55)5det(;33)3det(

• 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 it’s determinant, or a number, in which case it’s absolute value)

• 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

– 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; what’s left is a matrix of order 2-1=1

!

2221

1211 onethis

aa

aa

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

22#

#*

a

222211 aaA

– 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 1+1=2; the usual notation for cofactor is c with original indices

221111

11 )1( aAc

– Take now the second entry and build its cofactor: remove the second column and the first row; what’s left is a 1x1 matrix, namely

– compute the minor

2121 #

*#a

a

2112 aA

– Get the cofactor:

– we exhausted the row, and now we construct the sum:

211221

12 )1( aAc

2112221112121111

2221

1211

aaaacaca

aa

aa

• 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:

1715235)2(123

51

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

333231

232221

131211

aaa

aaa

aaa

– Choose a row - again, usually we take the first row, so let’s 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:

3332

2322

#

#

##*

aa

aa

– We compute this smaller (2x2) matrix’ determinant (because we know now how!) and so we get the minor of index 11:

– finally, the cofactor:

322333223332

232211 aaaa

aa

aaA

)()1( 322333221111

11 aaaaAc

– Next entry’s cofactor: (less talk, just computation)

3331

2321

3331

2321

#

#

#*#

aa

aa

aa

aa

312333213331

232112 aaaa

aa

aaA

– So, the cofactor is:

– and, for the last entry of this row, voila the cofactor:

)()1( 312333211221

12 aaaaAc

)()1()1( 312232213231

2221413

3113 aaaa

aa

aaAc

– Hence the determinant for our 3x3 matrix is

)()()( 312232211331233321123223332211

131312121111

333231

232221

131211

aaaaaaaaaaaaaaa

cacaca

aaa

aaa

aaa

• 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)

• Example:

0

4872105968445

186942753

384762951

987

654

321

• 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)

• 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 don’t need to compute those cofactors!

• 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 entry’s cofactor!

334

040

321

– the determinant is hence (I‘m mentioning the other 2 cofactors, but, as you see, they get multiplied by 0, so we don’t care what values they have):

912334

31

34

31

3#4

#*#

3#1

22

A

36)9(4)1(4040

334

040

321

2222

232221 Accc

• 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 a’s)

• 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

• 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?)

– 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

0

512

321

321

2211

200

210

321

– 4. If one modifies the matrix by adding a multiple of one row to another row (same for columns), the determinant doesn’t change - here you see the connection to elementary matrix operations!

21)103(312

53)1(3

123

003

531

123

553314

531

123

534

531

12

– 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

21

43

43

21

53

213

53

2313

– 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)

144

62

72

312

72

312 2

– 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

BABA

AAT

• One last thing: as you can see, it’s 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) that’s exactly what we get

225

313

401

234

210

531

• So … why not use this? As you can see, one elementary operation doesn’t 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.)

• Here’s an example:

– think of this as “factoring out the 3 out of the second row”

184

321

032

3

184

963

032

1100

670

321

3

184

032

321

3

184

321

032

312

13

2

4

RR

RR

– (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!)

231))11()7(1(3

• It’s especially useful for higher order matrices (4x4, 5x5, etc) since we don’t 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 …)