3.9.3 A trick for calculating determinants
a
b
• Consider the 2x2 matrix A =
c d
• Add the second column to the first and calculate
the determinant:
a+b b
c+d d
= (a+b)d –b(c+d)
= (ad+bd) –(bc+bd)
= (ad–bc)
=|A|
3.9.3 A trick for calculating determinants
• In fact if you replace any column of a matrix by
the original column + a multiple of any other
column the determinant is unchanged.
• Similarly, if you replace any row of a matrix by
the original row + a multiple of any other row
the determinant is unchanged.
• WARNING: adding one row or column to itself
will in general change the determinant
• Example:
1
A= a
1
b
1
c
C2
b+c a+c a+b
• So, using the top row:
| A | = (b-a)(a-c) – (c-a)(a-b)
= ba-a2+ac-bc
-(ac-bc -a2+ab)
=0
C2-C1
1
a
0 1
b-a c
b+c a-b a+b
C3
1
a
C3-C1
0
0
b-a c-a
b+c a-b a-c
3.9.4 More determinant properties
• If we take the transpose of a matrix, its
determinant is unchanged: |A| = |AT|
• For diagonal or upper triangular or lower
triangular matrices, the determinant is the
product of the leading diagonal entries:
a11 0 0
a11 a12 a13 a11 0 0
0 a22 0 = 0 a22 a23 = a21 a22 0 = a11a22a33
0 0 a33
0 0 a33 a31 a32 a33
3.9.4 More determinant properties
• Multiplying a whole row (or column) by k
multiplies the determinant by k.
• If a matrix is nxn then multiplying the matrix by k
is the same as multiplying n rows by k. Hence, the
determinant is multiplied by kn.
ka kb
a b
=
k
c d
c d
ka kb
a
b
2
=
k
kc kd
c d
3.9.4 More determinant properties
• If we swap two rows (or two columns), the
determinant changes by a factor of (-1):
c d
a b
a b
= (-1) c d
• If an entire row or column is zero, the
determinant is zero
1
2
0
0 -1
-1 0 7
4 5 = 3 0 5 =0
0 0
1 0 4
3.9.4 More determinant properties
• The determinant of a product is the product of
determinants:
|A B| = |A| |B|
• Example
1
2
A=
3 4
-1
2
AB=
-1 6
1
2
B=
-1 0
|A| = -2, |B| = 2
So, |A B| = -4 = (-2)(2) = |A||B|
3.9.5 Cross product as determinant
a1
• Consider two vectors: a = a2
a3
• Cross product is given by
b1
b = b2
b3
i j k
a x b = a 1 a2 a3
b 1 b2 b3
• Where i, j and k are unit vectors in the x,y and z
directions.
• Notice that a x b = - b x a