5- Determinants
2 by 2 Determinant
If a, b, c, and c are four real numbers , the symbol
is called a 2 by 2 determinant. Its value is the real
number ad – bc.
Then
Example:
3 by 3 Determinant
A 3 by 3 de terminant is symbolize by
indicates: it is in row i, column j.
Its value is given by
each 2 by 2 determinant is obtained after removing the row
and column that contain 
Example:
Evaluate
6 Complex Numbers
When we discussed set of numbers, we saw that the equation 3x = 1 did not have a
solution in the set of integers since there is no integer number that multiply
to 3 is equal
to 1. For that reason the set of rational numbers was created and the number 1/3
is a
solution of the equation.
Now. consider the equation x2 = -1. Since the square of any real number is
nonnegative,
then there is no real number that satisfies that equation. To remediate this
situation we
will create a new set of numbers such that when we square them, they will
produce
negative real numbers. This new set of numbers is called imaginary numbers.
Definition: Imaginary unit, i
The imaginary unit is denoted by the letter i and it is defined as
where i2 = -1. |
Remark:
Notice that the fourth power of i is 1. This is telling us
that the powers of i are cyclical,
with a period equal 4.
Example: Simplify the Imaginary Unit raised to Power
since 110 ÷ 4 = (27)·4 + 2
Example: Using Imaginary Numbers to Simplify Radicals
Definition: Any number in the form bi, where b is any real
number is an imaginary
number.
When a real number and an imaginary number are combined, the result is a complex
number.
Definition: Complex Numbers
A complex number z, in standard form is defined as |
z =a + bi
where a and b are real numbers and i is the imaginary unit. |
In the above definition, we call a the real part of the
complex number z and b is the
imaginary part of the complex number.
Complex numbers in the form z = bi are called pure imaginary numbers.
We dente by !, the set of all complex numbers.
Examples:
(a) 2 – 3i, 2 is the real part and the imaginary part is -3
(b) -5 + i , -5 is the real part and 1 is the imaginary part.
(c) –πi = 0 – πi, the real part is 0 and the imaginary part is –π.
(d) 7 = 7 + 0i, the real part is 7 and the imaginary part is 0.
(e) 0 = 0 + 0i, the real and the imaginary part are 0.
Remark: The set of real numbers is included in the set of complex numbers.
Definition: Equality of complex numbers
Two complex numbers 
and
are equal if and only if
a = c and b = d
Two complex numbers are equal if they have the same real part and the same
imaginary
part.
We can treat complex numbers z = a + bi, similar to binomials a + bx. So we add
and
subtract complex numbers as we add and subtract binomials , real parts are
combined with
real parts, and imaginary parts are combined to imaginary parts.
Example: Adding and Subtracting Complex Numbers
(a) (3 - 2i) + (-1 + i) = (3 – 1) + (-2 + 1)i = 2 + (-1)i = 2 – i
(b) (2 – i) – (3 – 4i) = 2 – i – 3 + 4i = (2 – 3) + (-1 + 4)i = -1 + 3i
To multiply complex numbers we apply all the methods of multiplying binomials.
It is
important to remember that i2 = -1.
Example Multiply the Complex Numbers
(a) (3 – i)(2 + i) = 3(2) + 3i – 2i – i(i) = 6 + i – (-1) = 7 + i
(b) 2i(-5 + i) = (2i)(-5) + (2i)(i) = -10i + 2(i2) = 2(-1) – 10i = -2 -10i
Definition: Complex Conjugates
Given the complex number z = a + bi, the complex number
is called the
complex conjugate of z and vice versa.
Remark: 
= (a + bi)(a – bi) = a2 –abi + abi – b2i2 = a2 + b2
When we divide two complex numbers to get the answer in
standard form, we multiply
and divide by the complex conjugate of the denominator.
Example: Dividing Complex Numbers
Quadratic Equations with a Negative Discriminant
Given the equation x2 = 9, it has two solutions x = 3 and x = -3.
Now, the equation x2 = -9 is solve
so, it has two solutions -3i and 3i.
Because we know to deal with the square root of a negative number, we can extend
the
quadratic formula to all cases of the discriminant..
Example: Solve the Quadratic Equation
x2 + 8 = 4x
Write the equation in standard form
x2 -4x + 8 = 0
a = 1, b = -4, c = 8
Applying quadratic formula
b2 – 4ac = (-4)2 – 4(8)(1) = 16 – 32 = -16 < 0,
The solution set is {2 – 2i, 2 + 2i}, they are complex
numbers
So, if the discriminant is negative, the quadratic equation has two unequal
solutions that
are conjugate complex numbers.
Character of the Solutions of a Quadratic Equation
In the complex number system consider the quadratic equation with real
coefficients
ax2 + bx + c = 0
(1) If b2 – 4ac > 0, the equation has two unequal real solutions
(2) If b2 – 4qc = 0 the equation has a repeated real solution, a double root
(3) If b2 – 4ac < 0, the equation has two complex solutions, they are not real
number.
They are the conjugate of each other.
Polar Form of a Complex Number
If P is a point in the complex plane corresponding to the complex number z = a +
bi that
can be viewed as the ordered-pair (a, b), then a = r cos ø and b = r sin ø,
where r =
is called the modulus or absolute value of
z, |z| and ø is called the argument of
z. It follows that z = a + bi = r(cos ø + i sin ø). This is the polar form of z.
The polar form of z is not unique, it is up to a multiple of 2πk, k an integer
z = r(cos(ø + 2πk) + i sin(ø + 2πk) , k any integer
De Moivre’s Theorem
Notice, there are n different n -th roots of the complex
number z.
Euler’s Formula
By as suming that the infinite series expansion
that we learned in calculus, holds when x = iø, we can get
this is called the Euler’s formula
Therefore, 
