3.9 Determinants. Given a square matrix A its determinant is a real number associated with the matrix. The determinant of A is written: det( A ) or  A  For a 2x2 matrix, the definition is. a. b. a. b. det = = ad  bc. c. d. c. d.
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det(A) or A
a
b
a
b
det = = ad  bc
c
d
c
d
2
det = = (5)(0) – (2)(2) = 4
2
0
Determinants 2x2 examples5
1
1
1
1
2
2
2
2
2
det = = (1)(4) – (2)(3) = 2
2
3
2
2
3
4
0
4
4
4
det = = (1)(4) – (2)(2) = 0
M11 : remove row 1, col 1
1
1
2
A =
1
2
3
2
3
M11 =
2
7
0
7
0
M12 : remove row 1, col 2
1
1
2
A =
1
2
3
1
3
M12 =
2
7
0
2
0
M13 : remove row 1, col 3
1
1
2
A =
1
2
3
1
2
M13 =
2
7
0
2
7
M21 : remove row 2, col 1
2
1
1
A =
1
2
3
1
2
M21 =
2
7
0
7
0
M22 : remove row 2, col 2
2
1
1
A =
1
2
3
1
2
M22 =
2
7
0
2
0
M23 : remove row 2, col 3
2
1
1
A =
1
2
3
1
1
M23 =
2
7
0
2
7
M31 : remove row 3, col 1
2
1
1
A =
1
2
3
1
2
M31 =
2
7
0
2
3
M32 : remove row 3, col 2
2
1
1
A =
1
2
3
1
2
M32 =
2
7
0
1
3
M33 : remove row 3, col 3
2
1
1
A =
1
2
3
1
1
M33 =
2
7
0
1
2
3.9.1 The formula for a 3x3 matrix
a13
a11
a12
a21
a23
A =
a22
a32
a31
a33
A = a11M11  a12M12 + a13M13
a22
a23
M11= = a22a33  a23a32
a32
a33
3.9.1 The formula for a 3x3 matrix
a13
a11
a12
a21
a23
A =
a22
a32
a31
a33
A = a11M11  a12M12 + a13M13
a21
a23
M12= = a21a33  a23a31
a31
a33
3.9.1 The formula for a 3x3 matrix
a13
a11
a12
a21
a23
A =
a22
a32
a31
a33
A = a11M11  a12M12 + a13M13
a21
a22
M13= = a21a32  a31a22
a31
a32
A = 1xM11  1xM12 + (2)xM13
1
1
2
1
3
2
1
3
2
A= 1x  1x + (2)
A =
1
2
3
2
0
7
2
0
7
2
7
0
= 1x(21) 1x(6) +(2)x(11) = 7
B = 0xM11  1xM12 + 3xM13
0
1
3
5
1
3
5
1
3
B= 0x  1x + 3 x
B =
5
3
1
1
0
2
1
0
2
1
2
0
= 0x(2) 1x(1) +(3)x(13) = 38
3.9.1 The formula for a 3x3 matrix
a13
a11
a12
a21
a23
A =
a22
a32
a31
a33
A = a11M11  a12M12 + a13M13
3.9.1 The formula for a 3x3 matrix
a13
a11
a12
a21
a23
A =
a22
a32
a31
a33
A = a11M11  a12M12 + a13M13
A = a21M21 + a22M22  a23M23
A = a31M31  a32M32 + a33M33
3.9.1 The formula for a 3x3 matrix
A = a11M11  a12M12 + a13M13
+
= a21M21 + a22M22  a23M23
= a31M31  a32M32 + a33M33

+


+
+

+
3.9.1 The formula for a 3x3 matrix
a13
+
a11
a12
a21
a23
A =
a22
a32
a31
a33

+
A = a11M11  a21M21 + a31M31


+
+

+
1
1
2
0
3
2
0
3
2
A= 1x  1x + (2) x
A =
0
2
3
0
1
1
0
1
1
0
1
1
= 1x(1) 1x(0) + (2)x(0) = 1
1
1
2
2
3
A= 1x  0 + 0 = 1x(1) = 1
A =
0
2
3
1
1
0
1
1
3.9.2 A general formula for determinants determinants. e.g.

+

+


+
+
+

+



+
+
3.9.2 A general formula for determinants determinants. e.g.
Cij:= (1)i+jMij
A= aijCijfor any j=1,2,...,n
A= aijCijfor any i=1,2,...,n