4 | The determinant
4 | The determinant
This chapter of Linear Algebra by Dr JH Klopper is licensed under an Attribution-NonCommercial-NoDerivatives 4.0 International Licence available at http://creativecommons.org/licenses/by-nc-nd/4.0/?ref=chooser-v1 .
4.1 Introduction
4.1 Introduction
This chapter embarks on an exploration of the matrix determinant, a scalar value that encapsulates significant information about a matrix, including its invertibility and the volume of the geometric figure it represents.
We start the journey with an exploration of minor entries, the determinants of the submatrices obtained by removing a row and a column from the original matrix. Understanding minor entries lays the foundation for comprehending the intricate structure of matrices and is a stepping stone towards calculating the determinant of larger matrices.
Building upon the concept of minor entries, we introduce cofactors, which are closely related to minor entries but involve an additional step of assigning a sign based on the position of the entry in the matrix. The section on cofactors unveils the role they play in the calculation of the determinant and their significance in the adjoint of a matrix.
Next, we delve into the effects of elementary row operations on the determinant. This section shows how various row operations, such as row interchange and scaling, influence the value of the determinant.
At the intersection of combinatorics and linear algebra, we explore the methods of using combinatorial techniques to calculate the determinant.
Then we venture into the realm of adjoint matrices, which are formed using the cofactors of the original matrix. This section elucidates the properties and applications of adjoint matrices, including their role in finding the inverse of a matrix.
Finally, we consider Cramer’s rule, a theorem that provides a method to solve linear systems using determinants.
4.2 Calculating a determinant
4.2 Calculating a determinant
Definition 4.2.1 The determinant is a scalar value that is derived from the elements of a square matrix. It has several important properties and plays a crucial role in linear algebra, particularly in the areas of matrix invertibility, linear transformations, and solution of systems of linear equations.
4.2.1 Calculating the determinant of a 2×2 matrix
4.2.1 Calculating the determinant of a 2×2 matrix
The notation and the calculation for the determinant of a matrix is shown in (1), where , such that is the real numbers or the complex numbers.
2×2
a,b,c,d∈
A=
det(A)=|A|=
=ad-bc
a | b |
c | d |
a | b |
c | d |
(
1
)
Problem 4.2.1.1 Calculate the determinant of the square matrix in (2).
The matrix and the solution is shown below.
A=
|A|=(3)(2)-(2)(1)=6-2=4
3 | 2 |
1 | 2 |
(
2
)The Det function calculate the determinant.
Det[{{3,2},{1,2}}](*Calculatingthedeterminantofthematrixrepresentedasanestedlistobejct*)
Out[]=
4
Problem 4.2.1.2 Calculate the determinant of the matrix generated by the code below and assigned to the variable matrixA.
matrixA={{1,},{1,2}};(*Creatingthematrixasanestedlistobject*)MatrixForm[matrixA](*Displayingthematrixinmathematicalform*)
Out[]//MatrixForm=
1 | |
1 | 2 |
Det[{{1,},{1,2}}](*CalculatingthedeterminantofmatrixA*)
Out[]=
2-
It should be clear from (1) this that some matrices could have a determinant of . We look at the consequences of this to the invertibility of a matrix next.
0
4.2.2 Using the determinant to calculate the inverse of a 2×2 matrix
4.2.2 Using the determinant to calculate the inverse of a matrix
2×2
Definition 4.2.2.1 The inverse of a matrix is calculated as shown in (3).
2×2
A=
=
a | b |
c | d |
-1
A
1
|A|
d | -b |
-c | a |
(
3
)A matrix is invertible if its determinant is not 0.
A
Problem 4.2.2.1 Calculate the inverse of the matrix below using the determinant.
The matrix and solution is shown in (4).
A=
=
=
=
4 | -2 |
3 | 6 |
-1
A
1
24-(-6)
6 | 2 |
-3 | 4 |
-1
A
6 30 | 2 30 |
-3 30 | 4 30 |
-1
A
1 5 | 1 15 |
-1 10 | 2 15 |
(
4
)We can evaluate our solution using the Inverse function.
In[]:=
MatrixForm[Inverse[{{4,-2},{3,6}}]]
Out[]//MatrixForm=
1 5 | 1 15 |
- 1 10 | 2 15 |
Problem 4.2.2.2 Show that the matrix below does not have an inverse by calculating the determinant.
The matrix and the solution is shown in (5).
A=
|A|=0=
1 | -1 |
-2 | 2 |
-1
A
1
0
2 | 1 |
2 | 1 |
(
5
)
4.2.3 Minor entries of a matrix
4.2.3 Minor entries of a matrix
Definition 4.2.3.1 The minor entries of a square matrix , denoted by , is the determinant of the submatrix after removing the row and the column .
A
n
M
ij
(n-1)×(n-1)
i
j
We create a matrix in (6) and calculate the determinant , that is, after removing row one and column one.
M
11
A=
=
=45-48=-3
1 | 2 | 3 |
4 | 5 | 6 |
7 | 8 | 9 |
M
11
5 | 6 |
8 | 9 |
(
6
)4.2.4 Cofactors of a matrix
4.2.4 Cofactors of a matrix
Definition 4.2.4.1 The cofactor of a square matrix is denoted as . Here we consider row and column and calculate as in (7).
A
n
C
ij
i
j
C
ij
C
ij
i+j
(-1)
M
ij
(
7
)So, for of above, we have the following, shown in (8).
C
11
A
C
11
1+1
(-1)
(
8
)Note that when is odd, then =- and when it is even =.
i+j
C
ij
M
ij
C
ij
M
ij
4.2.5 Determinant of a 3×3 matrix using cofactors
4.2.5 Determinant of a matrix using cofactors
3×3
Definition 4.2.5.1 The determinant of a matrix, with is calculated using (9), where is any of the rows.
A
n
n>2
k
det(A)=
n
∑
j=1
a
kj
C
kj
(
9
)We create a matrix below and calculate its determinant along .
3×3
k=1
In[]:=
matrixA={{3,1,2},{-1,2,0},{1,0,2}};MatrixForm[matrixA]
Out[]//MatrixForm=
3 | 1 | 2 |
-1 | 2 | 0 |
1 | 0 | 2 |
The values are shown in (10).
C
1j
C
11
1+1
(-1)
C
12
1+2
(-1)
C
13
1+3
(-1)
(
10
)The determinant is shown in (11).
3(4)+1(2)+2(-2)=10
(
11
)We use the Det function to verify the result.
In[]:=
Det[matrixA]
Out[]=
10
Definition 4.2.5.2 Cofactor expansion can also be done along columns. This is shown in (12), where is an arbitrary column.
k
4.2.6 Determinants of diagonal matrices
4.2.6 Determinants of diagonal matrices
Problem 4.2.6.1 Calculate the determinant of the square matrix generated in the code cell below and assigned to the variable matrixA.
4.3 Effect of elementary row operations on determinants
4.3 Effect of elementary row operations on determinants
Now we interchange rows one and three.
Definition 4.3.2 Interchanging any two rows has the effect of multiplying the determinant by -1.
Lastly we add 3 times row one to row three.
Definition 4.3.3 Adding a constant multiple of one row to another has no effect on the determinant.
4.4 Using combinatorics to calculate determinants
4.4 Using combinatorics to calculate determinants
The use of permutations to calculate the determinant of a square matrix is a method grounded in combinatorial mathematics. It is particularly useful for understanding the determinant from a more theoretical perspective.
Definition 4.4.32The associated permutations of an elementary product are the permutation defined by the columns of the elementary product.
The permutations of a square matrix takes a single element from each row, but as mentioned, for a specific permutation, no element can be in the same column. As example is shown in (14).
Definition 4.4.5 The signed elementary products are the products of the signs (-1 for odd and +1 for even) and the elementary products.
Problem 4.4.1 Calculate the values of the determinant of the matrix in (15) using permutations.
The associated permutations are listed in (16).
The elementary products are calculated in (17).
The inversions are shown in (18) for each associated permutation above.
The signed elementary products are calculated in (19).
The sum of the six signed products are calculated below.
4.5 Adjoint matrices
4.5 Adjoint matrices
4.5.1 Introducing adjoint matrices
4.5.1 Introducing adjoint matrices
4.5.2 Using adjoint matrices to calculate inverse
4.5.2 Using adjoint matrices to calculate inverse
We can now calculate the inverse, shown in (23).
Let’s check our work.
4.6 Cramer’s rule
4.6 Cramer’s rule
Problem 4.6.1 Solve the linear system shown in (25).
We start below, by creating a matrix of coefficients assigned to the variable matrixA.
Below, we use the Det function to calculate the determinant of the matrix of coefficients and assign the result to the variable detA.
4.7 Properties of the determinant
4.7 Properties of the determinant
4.8 Geometric interpretation of the determinant
4.8 Geometric interpretation of the determinant
The linearly transformed basis vectors and the parallelogram is shown in Figure 4.8.1.
The base of the parallelogram is calculated below using the Pythagorean theorem.
The height of the parallelogram (in the dotted blue line is calculated below.
The area is calculated below.
This is the determinant.