This document discusses matrices and determinants. It provides the general form of a matrix and defines what a determinant is. It then provides examples of how to calculate the determinant of matrices of different sizes (second order, third order, and higher). It also lists some theorems regarding how changing elements or rows/columns of a matrix affects its determinant value.

1. Matrices and Determinants
General Form:
a11 a12 a13 ... a1n
a21 a22 a23 ... a2n
. . . .
. . . .
. . . .
am1 am2 am3 . . . amn
Such an array arrange in rows and columns is called a matrix of size , or by .
For every matrix of order , for every matrix A, the corresponding value of the determinant
function is denoted by either |A| or det A. We call |A|, the determinant of A. Determinants are
numbers associated with a square matrix.
1. Second order matrix, n = 2
a11 a12
A=
a21 a22
To get the determinant of A
|A| = (a11)(a22) - (a12)(a21)
2. Third order matrix, n = 3
a11 a12 a13
A= a21 a22 a23
a31 a32 a33
To get the determinant of A
|A| = [(a11)(a22)(a33) + (a12)(a23)(a31) + (a13)(a21)(a32)] - [(a12)(a21)(a33) + (a11)(a23)(a32) + (a13)(a22)(a13)]
3. If n > 3
n
Then
|A| = a11A11 + a12A12 + … + a1nA1n = Σa
k=1
1kA1k
Where:
a1j = element in the first row
A1n = cofactor of a1j = (-1)1+j M1j
M1j = minor of a1j
Face Your Weaknesses
The best way to get better at anything and to be successful is to face the weaknesses we all possess. Everyone has weaknesses and in
order to be better, think clear, act appropriately, and succeed, you have to identify the areas you need to improve on and then take action
to turn your weaknesses into strengths.

2. Example: Find the determinant of A
2 1 1 2
1 2 2 2
A=
2 -1 -2 1
2 -2 1 -1
Solution:
|A| = (2)A11 + (1)A12 + (1)A13 + (2) A14
2 2 2
A11 = (-1)1+1 -1 -2 1 = -14
-2 1 -1
1 2 2
A12 = (-1)1+2 2 -2 1 = -21
2 1 -1
1 2 2
A13 = (-1)1+3 2 -1 1 =7
2 -2 -1
1 2 2
A14 = (-1)1+4 2 -1 -2 = 21
2 -2 1
|A| = (2)(-14) + (1)(-21) + (1)(7) + (2)(21)
|A| = 0
SEATWORK: Find the value of each of the following determinants.
(1.) (3.)
1 2 3 4 2 2 1 4
2 1 4 3 2 8 4 3
A= A=
3 4 2 1 5 4 11 13
4 3 1 2 4 6 1 2
(2.)
1 2 3 4 (4.)
9 2 7 6
4 3 2 1
A= 2 8 4 3
2 1 4 3 A=
1 2 8 4
3 4 1 2
3 6 9 2
Have Fun
When people start into the process of being successful, whether for personal growth or starting a business, they may start by incorporating
fun, but within a very short time, they realize it is hard work and the fun simply falls by the wayside.They enjoy life, the people around them,
and even find enjoyment in the challenges. This one element is often forgotten. This is a crucial element for success and should be a part of
your plan.

3. Theorems of Determinants
1. If two rows (columns) are interchanged, the value of the determinant is negative.
2 6 4
A= 5 2 3
3 1 5
|A| = (20 + 54 + 20) - (150 + 6 + 24) = - 86
3 1 5
A= 5 2 3
2 6 4
|A| = (24 + 6 + 150) - (20 + 54 + 20) = 86
2. If the element of a row (column) are all 0's, then |A| = 0
3 0 5
A= 5 0 3
2 0 4
|A| = 0
3. If one row (column) is proportional or equal to another row (column) then |A| = 0
2 6 4
A= 5 2 3
2 6 4
|A| = 0
4. If a constant k is multiplied to each element in a row (column), the value of the
determinant is multiplied by k.
2(2) 6(2) 4(2) 4 12 8
A= 5 2 3 = 5 2 3
3 1 5 3 1 5
|A| = - 172
Have Balance In Your Life
You have to find balance not only for yourself but also for others around you. Balance means providing time away from work for pleasure,
working extra hours when required, knowing when a new direction is required, etc.