1. MATRICES & DETERMINANTS
Monika V Sikand
Light and Life Laboratory
Department of Physics and Engineering physics
Stevens Institute of Technology
Hoboken, New Jersey,

2. OUTLINE Matrix Operations Multiplying Matrices
Determinants and Cramer’s Rule
Identity and Inverse Matrices
Solving systems using Inverse matrices

3. MATRIX A rectangular arrangement of numbers in rows and columns
For example:
2 rows
3 columns

4. TYPES OF MATRICES NAME DESCRIPTION EXAMPLE Row matrix
A matrix with only 1 row
Column matrix
A matrix with only I column
Square matrix
A matrix with same number of rows and columns
Zero matrix
A matrix with all zero entries

5. MATRIX OPERATIONS

6. COMPARING MATRICES
EQUAL MATRICES: Matrices having equal corresponding entries.
For Example:

7. ADDING MATRICES
Matrices of same dimension can be added
For Example:

8. SUBTRACTING MATRICES Matrices of same dimension can be subtracted
For example:

9. MULTIPLYING A MATRIX BY A SCALAR
For example:

10. SOLVING A MATRIX EQUATION
For example:

11. MULTIPLYING MATRICES

12. PRODUCT OF TWO MATRICES
For example:
FIND (a.) AB and (b.) BA

13. SOLUTION

14. SIMPLIFY
Simplify:
a.) A(B+C)
b.) AB+AC

15. SOLUTION
A(B+C):

16. SOLUTION
AB+AC:

17. DETERMINANTS & CRAMER”S RULE

18. DETERMINANT OF 22 MATRIX
The determinant of a 22 matrix is the difference of the entries on the diagonal.

19. EVALUATE
Find the determinant of the matrix:
Solution:

20. DETERMINANT OF 33 MATRIX
The determinant of a 33 matrix is the difference in the sum of the products in red from the sum of the products in black.
Determinant = [a(ei)+b(fg)+c(dh)]-[g(ec)+h(fa)+i(db)]

21. EVALUATE
Solution:

22. USING MATRICES IN REAL LIFE
The Bermuda Triangle is a large trianglular region in the Atlantic ocean. Many ships and airplanes have been lost in this region. The triangle is formed by imaginary lines connecting Bermuda, Puerto Rico, and Miami, Florida. Use a determinant to estimate the area of the Bermuda Triangle. N
Bermuda (938,454) . .
Miami (0,0) W E .
Puerto Rico (900,-518) S

23. SOLUTION
The approximate coordinates of the Bermuda Triangle’s three vertices are: (938,454), (900,-518), and (0,0). So the area of the region is as follows:
Hence, area of the Bermuda Triangle is about 447,000 square miles.

24. USING MATRICES IN REAL LIFE
The Golden Triangle is a large triangular region in the India.The Taj Mahal is one of the many wonders that lie within the boundaries of this triangle. The triangle is formed by the imaginary lines that connect the cities of New Delhi, Jaipur, and Agra. Use a determinant to estimate the area of the Golden Triangle. The coordinates given are measured in miles. . N
New Delhi (100,120) . .
Jaipur (0,0) W E
Agra (140,20) S

25. SOLUTION
The approximate coordinates of the Golden Triangle’s three vertices are: (100,120), (140,20), and (0,0). So the area of the region is as follows:
Hence, area of the Golden Triangle is about 7400 square miles.

26. USING MATRICES IN REAL LIFE
Black neck stilts are birds that live throughout Florida and surrounding areas but breed mostly in the triangular region shown on the map. Use a determinant to estimate the area of this region. The coordinates given are measured in miles. . N
(35,220) . .
(112,56) W E
(0,0) S

27. SOLUTION
The approximate coordinates of the Golden Triangle’s three vertices are: (35,220), (112,56), and (0,0). So the area of the region is as follows:
Hence, area of the region is about square miles.

28. CRAMER”S RULE FOR A 22 SYSTEM
Let A be the co-efficient matrix of the linear system: ax+by= e & cx+dy= f.
IF det A ≠0, then the system has exactly one solution. The solution is:
The numerators for x and y are the determinant of the matrices formed by using the column of constants as replacements for the coefficients of x and y, respectively.

29. EXAMPLE
Use cramer’s rule to solve this system:
8x+5y = 2
2x-4y = -10

30. SOLUTION Solution: Evaluate the determinant of the coefficient matrix
Apply cramer’s rule since the determinant is not zero.
The solution is (-1,2)

31. CRAMER”S RULE FOR A 33 SYSTEM
Let A be the co-efficient matrix of the linear system: ax+by+cz= j, dx+ey+fz= k, and gx+hy+iz=l.
IF det A ≠0, then the system has exactly one solution. The solution is:

32. EXAMPLE Compound Formula Atomic weight Methane CH4 16 Glycerol C3H8O3
The atomic weights of three compounds are shown. Use a linear system and Cramer’s rule to find the atomic weights of carbon(C ), hydrogen(H), and oxygen(O).
Compound
Formula
Atomic weight
Methane
CH4 16
Glycerol
C3H8O3 92
Water
H2O 18

33. SOLUTION Write a linear system using the formula for each compound
3C+ 8H + 3O = 92
2H + O =18
Evaluate the determinant of the coefficient matrix.

34. SOLUTION Apply cramer’s rule since determinant is not zero.
Atomic weight of carbon = 12
Atomic weight of hydrogen =1
Atomic weight of oxygen =16

35. IDENTITY AND INVERSE MATRICES

36. IDENTITY MATIX
22 IDENTITY MATRIX 3 IDENTITY MATRIX

37. INVERSE MATRIX
The inverse of the matrix

38. EXAMPLE
Find the inverse of
Solution:

39. CHECK THE SOLUTION

40. SOLVING SYSTEMS USING INVERSE MATRICES

41. SOLVING A LINEAR SYSTEM
-3x + 4y = 5
2x - y = -10
Writing the original matrix equation. A X B
AX = B
A-1AX = A-1B
IX = A-1B
X = A-1B

42. USING INVERSE MATRIX TO SOLVE THE LINEAR SYSTEM
-3x + 4y = 5
2x - y = -10
Hence the solution of the system is (-7,-4)