Determinants

• Theoretical Part
  
  • Determinant of a 1 x 1 matrix
    
  • Permutation of n ordered elements
    
  • Transposition
    
  • Theorem
    
  • Even and odd permutations
    
  • Sign of a permutation
    
  • The determinant of an n x n matrix
    
  • Row and columns of the determinant
    
  • Cofactor of an element a_i,j
    
  • A matrix A and its transpose have the same determinant.
    
  • Important result
    
  • Interchanging two columns of A
    
  • Multiplying a row of A by a real number
    
  • If A has two equal rows, |A| = 0
    
  • Addition of two determinants, which differ only from the i-th row
    
  • Equal determinants
    
  • The cofactor A_{1, 1}
    
  • The cofactor A_i,j
    
  • |I| = 1
    
  • Determinant of a diagonal matrix
    
  • Determinant of a product of two square matrices
    
• Quick Reference
  
  • Determinant of a 1x1 matrix
    
  • Determinant of a 2x2 matrix
    
  • Determinant of a 3x3 matrix
    
  • Cofactor of a_i,j
    
  • Determinant of a nxn matrix
    
  • Properties
    
  • Applying the properties
    
    • A few agreements on the data representation which is used here
      
    • Examples
      
  • Factorization of a determinant
    
• Next steps
  

Theoretical Part

With each square matrix corresponds just one number. This number is called the determinant of the matrix. The determinant of a matrix A is denoted det(A) or |A|. Now we'll define this correspondence.

Determinant of a 1 x 1 matrix

Permutation of n ordered elements

Transposition

Theorem

Even and odd permutations

Sign of a permutation

The determinant of an n x n matrix

 
 |A| =   sum     sgn(t) . a1,t(1) . a2,t(2) . a3,t(3) . ... . an,t(n)
       t in PS

Example1 : We want to calculate the determinant of a 2x2 matrix A.
Now n = 2 and there are only two permutations of S = (1,2).
t(1,2) = (1,2) with sgn(t) = +1
t'(1,2) = (2,1) with sign(t') = -1
We have only two terms +1.a_1,1 . a_2,2 and -1.a_1,2 . a_2,1
Thus the determinant of A is a_1,1 . a_2,2 - a_1,2 . a_2,1
We don't forget the rule :

 
|a  b|
|c  d|

= ad - cb

Example2 : We want to calculate the determinant of a 3x3 matrix A.
Now n = 3 and there are only 6 permutations of S = (1,2,3).
These 6 permutations transform (1, 2, 3) in:

 
(1, 2, 3) (2, 3, 1) (3, 1, 2)    (even permutations)
(3, 2, 1) (1, 3, 2) (2, 1, 3)    (odd permutations)

 
a1,1 . a2,2 . a3,3 + a1,2 . a2,3 . a3,1 + a1,3 . a2,1 . a3,2
-a1,3 . a2,2 . a3,1 - a1,1 . a2,3 . a3,2 - a1,2 . a2,1 . a3,3

 
|a  b  c|
|d  e  f|
|g  h  i|

 = aei + bfg + cdh - ceg - afh - bdi

To calculate larger determinants there are a lot of other methods involving various properties of determinants.

Row and columns of the determinant

Cofactor of an element a_i,j

Now consider A_i,j . a_i,j. This term uses the i-th row and the j-th column in the factor a_i,j. So the cofactor A_i,j is independent of the elements of the i-th row and the elements of the j-th column.
The cofactor A_i,j contains only elements from the matrix obtained from A by crossing out the i-th row and the j-th column.

Remark: If we write |A| = A_i,1 . a_i,1 + A_i,2 . a_i,2 + A_i,3 . a_i,3 + ... A_i,n . a_i,n, we say that the determinant is calculated emanating from the i-th row.
Example :

 
|a  b  c|
|d  e  f|
|g  h  i|

 = aei + bfg + cdh - ceg - afh - bdi

If we write |A| = (ch-bi)d + (ai-cg)e + (bg-ah)f , we say that the determinant is calculated emanating from the second row.
Similarly, we can start with a fixed column and then write |A| as a linear polynomial in a_{1, j}, a_2,j, a_3,j, ... a_n,j. Then one finds the same cofactors. So a_i,j has a unique cofactor A_i,j.

A matrix A and its transpose have the same determinant.

 
|B|  =  sum      sgn(t). b1,t(1) . b2,t(2) . b3,t(3) ... bn,t(n)
      t in PS

                   but bi,j = aj,i


     =    sum      sgn(t). at(1),1 . at(2),2 . at(3),3 ... at(n),n.
        t in PS

Then

 
 |B| =   sum         sgn(t') . a1,t'(1) . a2,t'(2) . a3,t'(3) ... an,t'(n)
      t' inverse of t

 Because  the set of all permutations of S  is the same set of all inverse permutations.
 So we can write

 |B| =    sum     sgn(t) . a1,t(1) . a2,t(2) . a3,t(3) ... an,t(n)
        t in PS

So, |B| = |A|.

Important result

Interchanging two columns of A

Sgn(t') = -1 and for each permutation t we have sign(t't) = -sign(t).

Say B is obtained by interchanging the column i and j of A.
For each k we have b_k,i = a_k,j and b_k,j = a_k,i or even for each k and each l we have

 
              bk,l  =  ak,t'(l)

 
|B| = sum     sgn(t). b1,t(1)   . b2,t(2)   . b3,t(3) ...   bn,t(n)
      t in PS

            but bk,l  =  ak,t'(l)

|B| = sum     sgn(t). a1,t't(1) . a2,t't(2) . a3,t't(3) ... an,t't(n)
     t in PS

|B| = sum    -sgn(t't). a1,t't(1) . a2,t't(2) . a3,t't(3) ... an,t't(n)
     t in PS

 
|B| = sum     -sgn(t"). a1,t"(1) . a2,t"(2) . a3,t"(3) ... an,t"(n)
     t" in PS

|B| = sum     -sgn(t) . a1,t(1)  . a2,t(2)  . a3,t(3) ...  an,t(n)
     t in PS

|B| =  -|A|

Multiplying a row of A by a real number

 
 = sum     sgn(t) . b1,t(1) . b2,t(2) . b3,t(3) ... bi,t(i) ... bn,t(n)
   t in PS

 = sum     sgn(t) . a1,t(1) . a2,t(2) . a3,t(3) ... r.ai,t(i) ... an,t(n)
   t in PS

 = sum     r sgn(t) . a1,t(1) . a2,t(2) . a3,t(3) ... ai,t(i) ... an,t(n)
   t in PS

 = r.(  sum    sgn(t) . a1,t(1) . a2,t(2) . a3,t(3) ... ai,t(i) ... an,t(n))
       t in PS

 = r.|A|

If A has two equal rows, |A| = 0

Addition of two determinants, which differ only from the i-th row

 
|A| = sum    sgn(t) . a1,t(1) . a2,t(2) . a3,t(3) ... ai,t(i) ... an,t(n)
     t in PS

|B| = sum    sgn(t) . b1,t(1) . b2,t(2) . b3,t(3) ... bi,t(i) ... bn,t(n)
     t in PS

   = sum     sgn(t) . a1,t(1) . a2,t(2) . a3,t(3) ... bi,t(i) ... an,t(n)
    t in PS

So |A| + |B|
   = sum  sgn(t) . a1,t(1) . a2,t(2) . a3,t(3) ... (ai,t(i) + bi,t(i)) ... an,t(n)
   t in PS

   =  determinant of the matrix formed by adding the i-th row from A and B,
    and taking the other elements from A or from B.

 
|a  b  c| |a  b'  c|   |a  b+b'  c|
|d  e  f|+|d  e'  f| = |d  e+e'  f|
|g  h  i| |g  h'  i|   |g  h+h'  i|

Equal determinants

 
            | a1,1   a1,2  ....
            | a2,1   a2,2  ....
            |
  |A| =     |
            | ai,1   ai,2  ....
            |
            |
            | aj,1   aj,2  ....
            |

 
             | a1,1   a1,2  ....
             | a2,1   a2,2  ....
             |
 |B| =       |
             | aj,1   aj,2  ....
             |
             |
             | aj,1   aj,2  ....
             |

 
             | a1,1   a1,2  ....
             | a2,1   a2,2  ....
             |
  |C| =      |
             |r.aj,1  r.aj,2  ....
             |
             |
             | aj,1   aj,2  ....
             |

 
             | a1,1   a1,2  ....
             | a2,1   a2,2  ....
             |
|A|+|C| =    |
             |aj,1+r.aj,1  aj,2+r.aj,2  ....
             |
             |
             | aj,1   aj,2  ....
             |

Therefore, a determinant does not change if we add a multiple of a row to another row. The same rule holds for columns
Ex.

 
|a  b  c|   |a+rd   b+re   c+rf|
|d  e  f| = | d      e      f  |
|g  h  i|   | g      h      i  |

The cofactor A_{1, 1}

 
|A| =  sum    sgn(t) . a1,t(1) . a2,t(2) . a3,t(3) ... an,t(n).
     t in PS

Now, instead of taking the sum for t in P_S we only take the sum for t in the set
{t in P_S | t(1) = 1} = permutations of S' =(2 ... n).

This sum then gives exactly A_1,1 . a_1,1.

 
A1,1 . a1,1

  =  sum    sgn(t) . a1,1 . a2,t(2) . a3,t(3) ... an,t(n)
   t in PS'

  =  a1,1 (  sum   sgn(t) a2,t(2) . a3,t(3) ... an,t(n) )
              t in PS'

Thus  A1,1 =

  sum   sgn(t) a2,t(2) . a3,t(3) ... an,t(n)
 t in PS'

Conclusion:
A_1,1 = the determinant of the sub-matrix of A obtained from A by crossing out the first row and the first column.

The cofactor A_i,j

Then we interchange in succession column j and j-1; j-1 and j-2; ... until e is on the first column and on the first row. This demands j-1 steps.

During this process the determinant of the matrix changes i+j-2 times sign.
Now the cofactor of e is the determinant of the sub-matrix of obtained from by crossing out the first row and the first column.

Now we return to the original matrix.
The value of A_i,j = (-1)^i+j-2.(the determinant of the sub-matrix of A, obtained from A by crossing out the i-th row and the j-th column).
Or stated simpler:
The value of A_{i, j} = (-1)^i+j.(the determinant of the sub-matrix of A, obtained from A by crossing out the i-th row and the j-th column).

|I| = 1

Let I be the (k+1) x (k+1) identity matrix and we calculate this matrix emanating from the first row.
|A| = A_1,1 . a_1,1 + A_1,2 . a_1,2 + A_1,3 . a_1,3 + ... A_1,n . a_1,n.
|A| = A_1,1.1 + A_1,2.0 + A_1,3.0 + ... A_1,n.0.
|A| = A_1,1
Now, the cofactor A_1,1 is the determinant of the k x k identity matrix, and this determinant is 1.

Determinant of a diagonal matrix

Determinant of a product of two square matrices

Quick Reference

Determinant of a 1x1 matrix

Determinant of a 2x2 matrix

 
|a  b|
|c  d|

 =  ad - cb

Determinant of a 3x3 matrix

 
|a  b  c|
|d  e  f|
|g  h  i|

 = aei + bfg + cdh - ceg - afh - bdi

Cofactor of a_i,j

Example

Take the 3 x 3 matrix A =

 
[4 5 7]
[1 2 3]
[2 5 6]

Determinant of a nxn matrix

|A| = A_i,1 . a_i,1 + A_i,2 . a_i,2 + A_i,3 . a_i,3 + ... A_i,n . a_i,n

A_{i, j} is the cofactor of a_i,j.
We say we have unfold the determinant following row i.

Example

Take the 3 x 3 matrix A =

 
[4 5 7]
[1 2 3]
[2 5 6]

We unfold the determinant following column 3.
The three cofactors are 1, -10, 3.
|A| = 7.1 - 3.10 + 6.3 = -5

Properties

• A matrix A and its transpose have the same determinant.
• If we swap two columns in A, |A| changes sign.
• If we swap two rows in A, |A| changes sign.
• If we multiply a row in A by a real number r, |A| changes in r.|A|
• If we multiply a column in A by a real number r, |A| changes in r.|A|
• If a row of a determinant only consists of zeros, the determinant is 0.
• If a column of a determinant only consists of zeros, the determinant is 0.
• If a determinant has two equal rows or two equal columns, the determinant is 0.
• If a determinant has two proportional rows or two proportional columns, the determinant is 0.

•  
  |a  b  c| |a  b'  c|   |a  b+b'  c|
  |d  e  f|+|d  e'  f| = |d  e+e'  f|
  |g  h  i| |g  h'  i|   |g  h+h'  i|
  

• A determinant does not change if we add a multiple of a row to another row.
• The same rule holds for columns
• The determinant of the identity matrix is 1.
• The determinant of a diagonal matrix is the product of the diagonal elements.
• |A|.|B| = |A.B|

Applying the properties

A few agreements on the data representation which is used here

 
R3  means  row 3
K2  means  column  2

R2 - R3 means:  replace  R2 by R2 - R3  (the value of the determinant does not change)

K2 + 2K3 means:  replace K2 by  K2 + 2.K3  (the value of the determinant does not change)

Examples

 
| x   2m   1 |
| 3   1    1 | =
| x   m    1 |

  R1 - R3       (create zeros)

| 0   m   0 |
| 3   1   1 | =
| x   m   1 |

 unfold the determinant following row 1

m.( -(3-x)) = m(x-3)

--------------------------------

| 1   2   3 |
| 4   5   6 | =
| 7   8   9 |

   R1 - R2 ; R2 - R3

| -3  -3  -3|
| -3  -3  -3| = 0
| 7   8   9 |

---------------------------------

| 1   1+m  -1 |
| 3   3+m  -3 | =
| 5    m   -1 |

   K1 + K3        (create zeros)

| 0   1+m  -1 |
| 0   3+m  -3 | =
| 4    m   -1 |

 unfold the determinant following column 1

0 + 0 + 4.((1+m).(-3) + (3+m))

= -8m

-------------------------------

| m2 + m    m    m3|
|   a        b    c  | =
|   d        e    f  |

    factor  m in the first row



  | m + 1    1    m2|
m.|   a      b    c  | =
  |   d      e    f  |

Factorization of a determinant

 
|ra  rb  rc|    |a  b  c|
|d    e   f| =r.|d  e  f|
|g    h   i|    |g  h  i|

Example

 
| a   a2   a3 |
| b   b2   b3 | =
| c   c2   c3 |

 factor a in  row 1
 factor b in  row 2
 factor c in  row 3


      | 1   a   a2 |
a.b.c.| 1   b   b2 | =
      | 1   c   c2 |

   R1 - R2 ; R2 - R3      (create zeros)

      | 0   a-b   a2-b2 |
a.b.c.| 0   b-c   b2-c2 | =
      | 1   c       c2   |

    factor in row 1
    factor in row 2


                  | 0   1   a+b |
a.b.c.(a-b).(b-c).| 0   1   b+c | =
                  | 1   c   c2 |

  unfold following the first column

a.b.c.(a-b).(b-c).(c-a)

Next steps

Systems of linear equations,
rank of a matrix,
Cramers Rule,
Classification of systems of linear equations,
Investigation of systems with a parameter