A matrix is a set of elements organized into rows and columns. Basic matrix operations include addition, subtraction, and multiplication. A matrix can be multiplied by another matrix if the number of columns of the first equals the number of rows of the second. The determinant of a matrix is a value that is used to determine properties of the matrix such as invertibility. Cramer's rule can be used to solve systems of linear equations involving matrices.

1. What is a Matrix?
• A matrix is a set of elements, organized into
rows and columns
m×n matrix is a matrix of m rows and n columns with
an order (mxn).
ù
úû
é
êë
n columns
a a
00 01
a a
10 11
m rows

2. Basic Operations
• Addition and Subtraction
ù
úû
a e b f
é
+ +
êë
+ +
ù
= úû
é
+ úû
êë
ù
é
êë
c g d h
e f
g h
a b
c d
ù
úû
a e b f
é
- -
êë
- -
ù
= úû
êë é
ù
- úû
é
êë
c g d h
e f
g h
a b
c d
Just add elements
Just subtract elements
A

3. Matrix aarriitthhmmeettiicc ((ooppeerraattiioonnss))
Matrix Addition/Subtraction
MMaattrriixx aaddddiittiioonn.. AAmm´nn aanndd BBmm´nn
• mmuusstt hhaavvee tthhee ssaammee nnuummbbeerrss ooff rroowwss aanndd
ccoolluummnnss
• aadddd ccoorrrreessppoonnddiinngg eennttrriieess
AAmm´nn ++ BBmm´nn == CCmm´nn == [[aaii,,jj ++ bbii,,jj]]
ù
ú ú ú
û
1 1
é
=
1 3
ê ê ê
ë
0 2
3,2 A
ù
ú ú ú
û
é
ê ê ê
ë
4 5
1 6
-
= -
2 3
3,2 B
ù
ú ú ú
û
é
ê ê ê
5 6
+ = -
ë
1 8
3 0
3,2 3,2 A B
MMaattrriixx ssuubbttrraaccttiioonn iiss ddoonnee ssiimmiillaarrllyy

4. Basic Operations
• Multiplication
ù
úû
ae bg af bh
é
êë
+ +
+ +
ù
= úû
é
êë
ù
úû
a b
é
êë
ce dg cf dh
e f
g h
c d
Multiply each row
by each column
An m×n can be multiplied by an n×p
matrix to yield an m×p result

5. Matrix Multiplication
Matrix aarriitthhmmeettiicc ((ooppeerraattiioonnss))
EExxaammppllee
ù
ú ú ú
û
1 0 4
é
=
0 2 2
ê ê ê
2 1 1
3 1 0
ë
4,3 A
ú úû ù
2 4
é
=
1 1
3 0
ê êë
3,2 B
ù
ú ú ú
14 4
8 9
7 13
û
é
ê ê ê
ë
4,2 AB C
= =
8 2
1 2 0 1 4 3 14 1,1 1,1 1,1 1,2 2,1 1,3 3,1 c = a b + a b + a b = × + × + × =
1 4 0 1 4 0 4 1,2 1,1 1,2 1,2 2,2 1,3 3,2 c = a b + a b + a b = × + × + × =
2 2 1 1 1 3 8 2,1 2,1 1,1 2,2 2,1 2,3 3,1 c = a b + a b + a b = × + × + × =

6. Basic Operations
• Transpose: Swap rows with columns
ù
ú ú ú
û
a b c
é
=
g h i
ê ê ê
ë
d e f
M
ù
ú ú ú
û
a d g
é
=
c f i
ê ê ê
ë
b e h
MT
ù
ú ú ú
V V T = [x y z]
û
x
é
=
z
ê ê ê
ë
y

7. Square Matrices
Square Matrices
• A Square matrix has same number of rows
and columns.
This is a 3x3 matrix

8. Row and Column Matrices
• A matrix can have single row (a “row
matrix”) or just a single column(a”column
matrix”)

9. Identity Matrix
Identity matrix: Square matrix with 1’s on the diagonal
ù
úû
é
0 1
êë
1 0
and zeros everywhere else
2 x 2 identity matrix
ù
ú ú ú
û
é
ê ê ê
ë
1 0 0
0 1 0
0 0 1
3 x 3 identity matrix
The identity matrix is to matrix multiplication as
__1_ is to regular multiplication!!!!

10. Multiply:
ù
úû
é
0 1
êë
1 0
ù
úû
5 2
é -
3 4
êë
ù
5 2
é -
3 4
= úû
êë
úûù
é
0 1
êë
1 0
ù
úû
5 2
é -
3 4
êë
ù
5 2
é -
3 4
= úû
êë
So, the identity matrix multiplied by any matrix
lets the “any” matrix keep its identity!
Mathematically, IA = A and AI = A !!

11. Determinant
• The 2×2 matrix,
`
has determinant

12. Example of 2 X 2 matrix
3 2
-
5 4
Notice the different symbol:
the straight lines tell you to
find the determinant!!
(3 * 4) - (-5 * 2)
12 - (-10)
22
=
3 2
-
5 4
=
=

13. Example of 3X3 matrix
a b c
a b c
a b c
1 1 1
2 2 2
3 3 3
b c
a
b c
b c
= 2 2
1
– + 3 3
1 1
2
3 3
a
b c
b c
1 1
a
3
b c
2 2
= a1 ( b2c3 -b3c2 ) – ( ) + 2 1 3 3 1 a b c -b c ( ) 3 1 2 2 1 a b c -b c
= ( ) 1 2 3 3 2 a b c -b c + ( ) + 2 1 3 3 1 a (-1) b c -b c ( ) 3 1 2 2 1 a b c -b c
= 1 1 a A + 2 2 + a A 3 3 a A

14. 2 3
5
-1 4
-
5 1 2
3 2 3
8 1 4
-
-
=

15. 2 3
5
-1 4
-
5 1 2
3 2 3
8 1 4
-
-
=
-
1 2
( 3)
1 4
-
– -

16. 2 3
5
-1 4
-
5 1 2
3 2 3
8 1 4
-
-
=
-
1 2
( 3)
1 4
-
-
-
1 2
8
2 3
– +

17. 2 3
5
-1 4
-
5 1 2
3 2 3
8 1 4
-
-
=
-
1 2
( 3)
1 4
-
-
-
1 2
8
2 3
– +
= 5( 8 - (-3)) – (-3)( 4 - 2) + 8( 3- (-4))

18. 2 3
5
-1 4
-
5 1 2
3 2 3
8 1 4
-
-
=
-
1 2
( 3)
1 4
-
-
-
1 2
8
2 3
– +
= 5( 8 - (-3)) – (-3)( 4 - 2) + 8( 3- (-4))
= 55 – (-6) + 56

19. Example-
2 3
5
-1 4
-
5 1 2
3 2 3
8 1 4
-
-
=
-
1 2
( 3)
1 4
-
-
-
1 2
8
2 3
– +
= 5( 8 - (-3)) – (-3)( 4 - 2) + 8( 3- (-4))
= 55 – (-6) + 56
= 117

20. Cramer’s Rule
This rule can be used to calculate solutions of
where A is a square matrix.
Let A be an n x n matrix. The system of equations
has a unique solution if and only if .

21. Let Ak be the matrix obtained by replacing column k of A by
the column matrix B . Then

22. Example:
Solve the following equations:-
x + 3 y + 3z = 1;
x + 4y +3 z = 0;
x + 3y + 4z = 2;
Ans: x = 1, y = -1, z = 1

23. Example:
The ABC shipping company charges $2.90 for all
packages weighing less than or equal to 5 lbs, $5.20
for packages weighing more than 5 lbs and less than
10 lbs, and $8.00 for all packages weighing 10 lbs. or
more. The number of packages weighing 5 lbs or less
is 50% of the number of packages weighing 10 lbs or
more. One day shipping charges for 300 orders was
$1,508. Find the number of packages in each category.

24. Sol:
x = pkgs less or equal to 5 lbs
y = pkgs between 5 and 10 lbs
z = pkgs 10 lbs or more
x + y + z = 300
2.9x + 5.2y + 8z = 1508
x = .5z which changes to x - .5z = 0
D determinant
| 1.......1......1 |
|2.9....5.2....8 |
| 1.......0.... -.5|