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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
det = = ad - bcac
bd
• For larger matrices the definition is more complicated
ac
bd
13
24
det = = (1)(4) – (2)(3) = -213
24
-5-2
20
det = = (-5)(0) – (2)(-2) = 4-5-2
20
12
24
det = = (1)(4) – (2)(2) = 012
24
Determinants 2x2 examples
• To define det(A) for larger matrices, we will need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A
Determinants
-21 1-1 22 7
3 0
A =M11 =
27
30
M11 : remove row 1, col 1
-21 1-1 22 7
3 0
A =M12 =
-12
30
M12 : remove row 1, col 2
Determinants
• To define det(A) for larger matrices, we will need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A
-21 1-1 22 7
3 0
A =M13 =
-12
27
M13 : remove row 1, col 3
Determinants
• To define det(A) for larger matrices, we will need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A
-21 1-1 22 7
3 0
A =M21 =
17
-20
M21 : remove row 2, col 1
Determinants
• To define det(A) for larger matrices, we will need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A
-21 1-1 22 7
3 0
A =M22 =
12
-20
M22 : remove row 2, col 2
Determinants
• To define det(A) for larger matrices, we will need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A
-21 1-1 22 7
3 0
A =M23 =
12
17
M23 : remove row 2, col 3
Determinants
• To define det(A) for larger matrices, we will need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A
-21 1-1 22 7
3 0
A =M31 =
12
-23
M31 : remove row 3, col 1
Determinants
• To define det(A) for larger matrices, we will need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A
-21 1-1 22 7
3 0
A =M32 =
1-1
-23
M32 : remove row 3, col 2
Determinants
• To define det(A) for larger matrices, we will need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A
-21 1-1 22 7
3 0
A =M33 =
1-1
12
M33 : remove row 3, col 3
Determinants
• To define det(A) for larger matrices, we will need the definition of a minor Mij
• The minor Mij of a matrix A is the matrix formed by removing the ith row and the jth column of A
• For a matrix a11
A = a21
a12
a31
a22
a32
a13
a23
a33
• Its determinant is given by
|A| = a11|M11| - a12|M12| + a13|M13|
• From the formula for a 2x2 matrix:
|M11|= = a22a33 - a23a32 a22
a32
a23
a33
3.9.1 The formula for a 3x3 matrix
• For a matrix a11
A = a21
a12
a31
a22
a32
a13
a23
a33
• Its determinant is given by
|A| = a11|M11| - a12|M12| + a13|M13|
• From the formula for a 2x2 matrix:
|M12|= = a21a33 - a23a31 a21
a31
a23
a33
3.9.1 The formula for a 3x3 matrix
• For a matrix a11
A = a21
a12
a31
a22
a32
a13
a23
a33
• Its determinant is given by
|A| = a11|M11| - a12|M12| + a13|M13|
• From the formula for a 2x2 matrix:
|M13|= = a21a32 - a31a22 a21
a31
a22
a32
3.9.1 The formula for a 3x3 matrix
-21 1-1 22 7
3 0
A =
= 1x(-21) -1x(-6) +(-2)x(-11) = 7
|A|= 1x - 1x + (-2)27
30
-12
30
-12
27
|A| = 1x|M11| - 1x|M12| + (-2)x|M13|
3x3 Example
30 15 3-1 2
1 0
B =
= 0x(-2) -1x(1) +(3)x(13) = 38
|B|= 0x - 1x + 3 x32
10
5-1
10
5-1
32
|B| = 0x|M11| - 1x|M12| + 3x|M13|
3x3 Example
• For the matrix a11
A = a21
a12
a31
a22
a32
a13
a23
a33
• We used the top row to calculate the determinant:
|A| = a11|M11| - a12|M12| + a13|M13| • However, we could equally have used any row of
the matrix and performed a similar calculation
3.9.1 The formula for a 3x3 matrix
• For the matrix a11
A = a21
a12
a31
a22
a32
a13
a23
a33
• Using the top row:|A| = a11|M11| - a12|M12| + a13|M13|
• Using the second row |A| = -a21|M21| + a22|M22| - a23|M23|
• Using the third row
|A| = a31|M31| - a32|M32| + a33|M33|
3.9.1 The formula for a 3x3 matrix
|A| = a11|M11| - a12|M12| + a13|M13|
• Notice the changing signs depending on what row we use:
= -a21|M21| + a22|M22| - a23|M23|
= a31|M31| - a32|M32| + a33|M33|
+ +
++
+-
--
-
3.9.1 The formula for a 3x3 matrix
• Equally, we could have used any column as long as we follow the signs pattern
+ +
++
+-
--
-
• E.g. using the first column:
|A| = a11|M11| - a21|M21| + a31|M31|
a11
A = a21
a12
a31
a22
a32
a13
a23
a33
3.9.1 The formula for a 3x3 matrix
• This choice sometimes makes it a bit easier to calculate determinants. e.g.
-21 1 0 20 1
3 1
A =
|A|= 1x - 1x + (-2) x21
31
0 0
31
0 0
21
= 1x(-1) -1x(0) + (-2)x(0) = -1
• Using the first row:
• This choice sometimes makes it a bit easier to calculate determinants. e.g.
-21 1 0 20 1
3 1
A =
|A|= 1x - 0 + 0 = 1x(-1) = -1 21
31
• However, using the first column:
• For a 4x4 matrix we add up minors like the 3x3 case, and again use the same signs pattern
+ +
++
+-
--
-+ +
+
- --
-
• Notice that if we think of the signs pattern as a matrix, then it can be written as (-1)i+j
3.9.2 A general formula for determinants
• For a nxn matrix A=(aij) the co-factors of A are defined by
Cij:= (-1)i+j|Mij|• The determinant of A is given by the formula
|A|= aij Cij for any j=1,2,...,n
n
i 1• Or,
|A|= aij Cij for any i=1,2,...,n
n
j 1
3.9.2 A general formula for determinants