Chapters
Determinants
Suppose A is an n x n matrix. Associated with A is a number called the determinant of A and is denoted by det A or |A|. Symbolically, we distinguish a matrix A from the determinant of A by replacing the parentheses by vertical bars:
[ A =begin{bmatrix}
a_{11} & a_{12}& dots & a_{1n}
a_{21} & a_{22}& dots & a_{2n}
vdots & & & vdots
a_{n1} & a_{n2}& dots & a_{nn}
end{bmatrix}]
and
[det A =begin{vmatrix}
a_{11} & a_{12}& dots & a_{1n}
a_{21} & a_{22}& dots & a_{2n}
vdots & & & vdots
a_{n1} & a_{n2}& dots & a_{nn}
end{vmatrix}]
Thus, every square matrix A, is assigned a particular scalar quantity called the determinant of A, denoted by det A or |A|.
A determinant of an n x n matrix is said to be a determinant of order n.
Let's begin by defining the determinants of 1 x 1, 2 x 2 , and 3 x 3 matrices.
The Determinant of Order One
For a 1 x 1 matrix A = (a) we have det A = |a| = a.
For example, if A = (-5), then det A = |-5| = -5.
Note: In this case, the vertical bars || around a number do not mean the absolute value of the number.
The Determinant of Order Two
The determinant of a 2 x 2 matrix is said to be a determinant of order 2.
Consider matrix A:
[ A =begin{bmatrix}
a_{11} & a_{12}
a_{21} & a_{22}
end{bmatrix}]
The determinant of
[ A =begin{bmatrix}
a_{11} & a_{12}
a_{21} & a_{22}
end{bmatrix}]
is the number
[det A =begin{vmatrix}
a_{11} & a_{12}
a_{21} & a_{22}
end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}]
As a mnemonic, a determinant of order 2 is thought to be the product of the main diagonal entries of A minus the product of the other diagonal entries.
For example, if matrix A is the 2 x 2 matrix
[ A =begin{bmatrix}
5 & -7
2 & 9
end{bmatrix}],
then det A is
[begin{vmatrix} A = end{vmatrix} begin{vmatrix}
a_{11} & a_{12}
a_{21} & a_{22}
end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}]
The Determinant of Order Three
The determinant of a 3 x 3 matrix is said to be a determinant of order 3.
[ A =begin{bmatrix}
a_{11} & a_{12} & a_{13}
a_{21} & a_{22} & a_{23}
a_{31} & a_{32} & a_{33}
end{bmatrix}]
The determinant of
[ A =begin{bmatrix}
a_{11} & a_{12} & a_{13}
a_{21} & a_{22} & a_{23}
a_{31} & a_{32} & a_{33}
end{bmatrix}]
is the number
[det A =begin{vmatrix}
a_{11} & a_{12} & a_{13}
a_{21} & a_{22} & a_{23}
a_{31} & a_{32} & a_{33}
end{vmatrix}]
[=a_{11}a_{22}a_{33}+a_{12}a_{23}a_{31}+a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}]
Note that there are six products, each consisting of three elements in the matrix.
Three of the products appear with a positive sign (they preserve their sign) and three with a negative sign (they change their sign).
The expression in (3) can be written in a more tractable form. By factoring, we have
[det A = a_{11}(a_{22}a_{33} - a_{23}a_{32}) + a_{12}(-a_{21}a_{33} + a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})]
But in view of (1), each term in parentheses is recognized as the determinant of a 2 x 2 matrix:
[det A = a_{11}begin{vmatrix}
a_{22} & a_{23}
a_{32} & a_{33}
end{vmatrix}
+ a_{12} begin{bmatrix} - begin{vmatrix} a_{21} & a_{23}
a_{31} & a_{33} end{vmatrix} end{bmatrix}]
a_{21} & a_{22}
a_{31} & a_{32}
end{vmatrix}].
[= a_{12} C_{12} + a_{22} C_{22} + a_{32} C_{32},]
which is the cofactor expansion of det A along the second column. It is left as an exercise to show from (3) that det A can also be expanded by cofactors along the third row:[det A = a_{31}C_{31} + a_{32}C_{32} + a_{33}C_{33}.]We are, of course, suggesting in (5), (8) and (9) the following general result:The determinant of a 3 x 3 matrix can be evaluated by expanding det A by expanding det A by cofactors along any row or along any column.
Cofactor Expansion Along the First Row
Cofactor Expansion Along the Third Column
Evaluate the determinant of [ A = begin{bmatrix} 6&5&0-1&8&-7 -2&4&0end{bmatrix}]
Solution: Since there are two zeros in the third column, we expand by cofactors of that column:
[det A = begin{vmatrix} 6&5&0-1&8&-7-2&4&0end{vmatrix} = 0C_{13} + (-7)C_{23} + 0C_{33}]
[=(-7)(-1)^{2+3} begin{vmatrix} 6&5&01&8&7-2&4&0end{vmatrix}=(-7)(-1)^{2+3}begin{vmatrix} 6&5-2&4end{vmatrix}]
[=7[6(4)-5(-2)]=238]
Carrying the above ideas one step further, we can evaluate the determinant of a 4 X 4 matrix by multiplying the entries in a row (or column) by their corresponding cofactors and adding the products. In this case, each cofactor is a signed minor determinant of an appropriate 3 X 3 submatrix. The following theorem, which we shall give without proof, states that the determinant of any n X n matrix A can be evaluated by means of cofactors.
Cofactor Expansion of a Determinant
[Let A =(a_{i}_{j})_{ntimes n}] be an [ntimes n] matrix.
For each 1 ≤ i ≤ n, the cofactor of det A along the ith row is:
[det A = a_{i}_{1}C_{i}_{1} + a_{i}_{2}C_{i}_{2} + ... + a_{i}_{n}C_{i}_{n}]
For each 1 ≤ j ≤ n, the cofactor expansion of det A along the jth column is:
[det A = a_{1}_{j}C_{1}_{j} + a_{2}_{j}C_{2}_{j} + ... + a_{n}_{j}C_{n}_{j}]
The sign factor pattern for the cofactors illustrated in (7) extends to matrices of order greater than 3.
. + - + - + ...
+ - + - - + - + - ...
- + - + + - + - + ...
+ - + - - + - + - ...
- + - + + - + - + ...
. : : : : :
4 X 4 matrix n X n matrix
Cofactor Expansion Along the Fourth Row
Evaluate the determinant of [ A = begin{bmatrix} 5&1&2&4-1&0&2&31&1&6&11&0&0&-4end{bmatrix}]
Solution: Since the matrix has two zero entries in its fourth row, we choose to expand det A by cofactors along that row:
[det A = begin{vmatrix} 5&1&2&4-1&0&2&31&1&6&11&0&0&-4end{vmatrix} = (1)C_{41} + (0)C_{42} + (0)C_{43}+ (-4)C_{44}]
(10)
where
[C_{41} = ( - 1)^{4+1} begin{vmatrix} 1&2&4 &2&3 1&6&1end{vmatrix}]
and
[C_{44}=( - 1) ^{4+4} begin{vmatrix} 5&1&2-1&0&21&1&6end{vmatrix}]
We then expand both these determinants by cofactors along the second row:
[C_{41}=( - 1)begin{vmatrix} 1&2&4 &2&31&6&1end{vmatrix}]
=
[-(0(-1) ^{2+1}begin{vmatrix} 2&46&1end{vmatrix}+2(-1)^{2+2}begin{vmatrix} 1&41&1end{vmatrix}+3(-1)^{2+3}begin{vmatrix} 1&21&6end{vmatrix}]
= 18
[C_{44}=begin{vmatrix} 5&1&2-1&0&21&1&6end{vmatrix}]
=
[(-1)(-1) ^{2+1}begin{vmatrix} 1&21&6end{vmatrix}+0(-1)^{2+2}begin{vmatrix} 5&21&6end{vmatrix}+2(-1)^{2+3}begin{vmatrix} 5&11&1end{vmatrix}]
= -4
Therefore (10) becomes
[det A = begin{vmatrix} 5&1&2&4-1&0&2&31&1&6&11&0&0&-4end{vmatrix} = (1)(18) + (-4)(-4) = 34]
You should verify this result by expanding det A by cofactors along the second column
Remarks
In previous mathematics courses you may have seen the following memory device, analogous to (2), for evaluating a determinant of order 3:
. multiply multiply (11)
[begin{vmatrix} a_{11} & a_{12} & a_{13}a_{21} & a_{22} & a_{23}a_{31} & a_{32} & a_{33}end{vmatrix}begin{vmatrix} a_{11} & a_{12}a_{21} & a_{22}a_{31} & a_{32}end{vmatrix}]
(i) Add the products of the entries on the arrows that go from left to right.
(ii) Subtract from the number in (i) the sum of the products of the entries on the arrows that go from right to left.
A word of caution is in order here. The memory device given in (11), though easily adapted to matrices larger than 3 X 3, does not give the correct results. There are no mnemonic devices for evaluating the determinants of order 4 or greater.
Complex Numbers
Definition
A complex number is any number of the form z = a + ib where a and b are real numbers and i is the imaginary unit. The imaginary unit i is defined by the property i² = -1. using the imaginary unit, a general complex number is build out of two real numbers.
Terminology
The number i in z = a + ib is called the imaginary unit. The real number x in z = x + iy is called the real part of z; the real number y is called the imaginary part of z. The real and imaginary parts of a complex number z are abbreviated Re(z) and Im(z), respectively. For example, if z = 4 -9i, then Re(z) = 4 and Im(z) = -9. A real constant multiple of the imaginary unit is called a pure imaginary number. For example, z = 6i is a pure imaginary number. Two complex numbers are equal if their real and imaginary parts are equal. Since this simple concept is sometimes useful, we formalize the last statement in the next definition.