1. 3.9 Determinants det(A) or |A| a b a b det = = ad - bc c d c d
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
For larger matrices the definition is more complicated

2. Determinants 2x2 examples1 3 2 4 1 3 2 4
det = = (1)(4) – (2)(3) = -2 -5 -2 2
det = = (-5)(0) – (2)(-2) = 4 1 2 4
det = = (1)(4) – (2)(2) = 0

3. Determinants M11 : remove row 1, col 1 -2 1 -1 2 7 3 A = 2 3 M11 = 7
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
M11 : remove row 1, col 1 -2 1 -1 2 7 3
A = 2 3
M11 = 7

4. Determinants M12 : remove row 1, col 2 -2 1 -1 2 7 3 A = -1 3 M12 = 2
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
M12 : remove row 1, col 2 -2 1 -1 2 7 3
A = -1 3
M12 = 2

5. Determinants M13 : remove row 1, col 3 -2 1 -1 2 7 3 A = -1 2 M13 = 2
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
M13 : remove row 1, col 3 -2 1 -1 2 7 3
A = -1 2
M13 = 2 7

6. Determinants M21 : remove row 2, col 1 1 -1 2 7 3 A = -2 1 -2 M21 = 7
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
M21 : remove row 2, col 1 1 -1 2 7 3
A = -2 1 -2
M21 = 7

7. Determinants M22 : remove row 2, col 2 1 -1 2 7 3 A = -2 1 -2 M22 = 2
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
M22 : remove row 2, col 2 1 -1 2 7 3
A = -2 1 -2
M22 = 2

8. Determinants M23 : remove row 2, col 3 1 -1 2 7 3 A = -2 1 1 M23 = 2 7
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
M23 : remove row 2, col 3 1 -1 2 7 3
A = -2 1 1
M23 = 2 7

9. Determinants M31 : remove row 3, col 1 1 -1 2 7 3 A = -2 1 -2 M31 = 2
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
M31 : remove row 3, col 1 1 -1 2 7 3
A = -2 1 -2
M31 = 2 3

10. Determinants M32 : remove row 3, col 2 1 -1 2 7 3 A = -2 1 -2 M32 = -1
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
M32 : remove row 3, col 2 1 -1 2 7 3
A = -2 1 -2
M32 = -1 3

11. Determinants M33 : remove row 3, col 3 1 -1 2 7 3 A = -2 1 1 M33 = -1
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
M33 : remove row 3, col 3 1 -1 2 7 3
A = -2 1 1
M33 = -1 2

12. 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:
|M11|= = a22a33 - a23a32
a22
a32
a23
a33

13. 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:
a21
a23
|M12|= = a21a33 - a23a31
a31
a33

14. 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:
a21
a22
|M13|= = a21a32 - a31a22
a31
a32

15. 3x3 Example -2 1 -1 2 7 3 A = |A| = 1x|M11| - 1x|M12| + (-2)x|M13|
A =
|A| = 1x|M11| - 1x|M12| + (-2)x|M13|
|A|= 1x x (-2) 2 7 3 -1
= 1x(-21) -1x(-6) +(-2)x(-11) = 7

16. 3x3 Example 3 1 5 -1 2 B = |B| = 0x|M11| - 1x|M12| + 3x|M13|1 5 -1 2
B =
|B| = 0x|M11| - 1x|M12| + 3x|M13|
|B|= 0x x x 3 2 1 5 -1
= 0x(-2) -1x(1) +(3)x(13) = 38

17. 3.9.1 The formula for a 3x3 matrix
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

18. 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|

19. 3.9.1 The formula for a 3x3 matrix
|A| = a11|M11| - a12|M12| + a13|M13|
= -a21|M21| + a22|M22| - a23|M23|
= a31|M31| - a32|M32| + a33|M33|
Notice the changing signs depending on what row we use: + -

20. 3.9.1 The formula for a 3x3 matrix
Equally, we could have used any column as long as we follow the signs pattern + -
a11
A =
a21
a12
a31
a22
a32
a13
a23
a33
E.g. using the first column:
|A| = a11|M11| - a21|M21| + a31|M31|

21. This choice sometimes makes it a bit easier to calculate determinants
This choice sometimes makes it a bit easier to calculate determinants. e.g. -2 1 2 3
A =
Using the first row:
|A|= 1x x (-2) x 2 1 3
= 1x(-1) -1x(0) + (-2)x(0) = -1

22. This choice sometimes makes it a bit easier to calculate determinants
This choice sometimes makes it a bit easier to calculate determinants. e.g. -2 1 2 3
A =
|A|= 1x = 1x(-1) = -1 2 1 3
However, using the first column:

23. 3.9.2 A general formula for determinants
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

24. 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
Or,
|A|= aij Cij for any i=1,2,...,n