1. Matrices

2. Element - Each value in a matrix; either a number or a constant.
Matrix - A rectangular array of variables or constants in horizontal rows and vertical columns enclosed in brackets.
Element - Each value in a matrix; either a number or a constant.
Dimension - Number of rows by number of columns of a matrix.
**A matrix is named by its dimensions.

3. Examples: Find the dimensions of each matrix.
Dimensions: 3x2
Dimensions: 4x1
Dimensions: 2x4

4. Different types of Matrices
Column Matrix - A matrix with only one column.
Row Matrix - A matrix with only one row.
Square Matrix - A matrix that has the same number of rows and columns.

5. 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

6. Equal Matrices - Two matrices that have the same dimensions and each element of one matrix is equal to the corresponding element of the other matrix.
*The definition of equal matrices can be used to find values when elements of the matrices are algebraic expressions.

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

8. Addition of Matrices
If A and B are both m × n matrices then the sum of A and B, denoted A + B, is a matrix obtained by adding corresponding elements of A and B.
If A and B are both m × n matrices then the sum of A and B, denoted A + B, is a matrix obtained by adding corresponding elements of A and B.
add these
add these
add these
add these
add these
add these

9. Matrix
addition is
commutative
Matrix
addition is
associative

10. Scalar Multiplication of Matrices
If A is an m × n matrix and s is a scalar, then we let kA denote the matrix obtained by multiplying every element of A by k. This procedure is called scalar multiplication.
PROPERTIES OF SCALAR MULTIPLICATION

11. The m × n zero matrix, denoted 0, is the m × n matrix whose elements are all zeros.
1 × 3
2 × 2

12. SOLVING A MATRIX EQUATION
For example:

13. Multiplication of Matrices
The multiplication of matrices is easier shown than put into words. You multiply the rows of the first matrix with the columns of the second adding products
Find AB
First we multiply across the first row and down the first column adding products. We put the answer in the first row, first column of the answer.

14. Find AB Notice the sizes of A and B and the size of the product AB.
Now we multiply across the second row and down the second column and we’ll put the answer in the second row, second column.
Now we multiply across the first row and down the second column and we’ll put the answer in the first row, second column.
Now we multiply across the second row and down the first column and we’ll put the answer in the second row, first column.
We multiplied across first row and down first column so we put the answer in the first row, first column.

15. To multiply matrices A and B look at their dimensions
MUST BE SAME
SIZE OF PRODUCT
If the number of columns of A does not equal the number of rows of B then the product AB is undefined.

16. 23 32 Now let’s look at the product BA. can multiply size of answer
across third row as we go down first column:
across third row as we go down second column:
across second row as we go down third column:
across second row as we go down second column:
across third row as we go down third column:
across second row as we go down first column:
across first row as we go down first column:
across first row as we go down second column:
across first row as we go down third column:
Completely different than AB!

17. PROPERTIES OF MATRIX MULTIPLICATION
Is it possible for AB = BA ?
,yes it is possible.

18. Multiplying a matrix by the identity gives the matrix back again.
What is AI?
What is IA?
identity matrix
an n  n matrix with ones on the main diagonal and zeros elsewhere

19. Can we find a matrix to multiply the first matrix by to get the identity?
Let A be an n n matrix. If there exists a matrix B such that AB = BA = I then we call this matrix the inverse of A and denote it A-1.

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

21. SOLUTION

22. INVERSE MATRIX
The inverse of the matrix

23. EXAMPLE
Find the inverse of
Solution:

24. CHECK THE SOLUTION

25. If A has an inverse we say that A is nonsingular
If A has an inverse we say that A is nonsingular. If A-1 does not exist we say A is singular.
To find the inverse of a matrix we put the matrix A, a line and then the identity matrix. We then perform row operations on matrix A to turn it into the identity. We carry the row operations across and the right hand side will turn into the inverse.
To find the inverse of a matrix we put the matrix A, a line and then the identity matrix. We then perform row operations on matrix A to turn it into the identity. We carry the row operations across and the right hand side will turn into the inverse.
r2
r1  r2
2r1+r2

26. Check this answer by multiplying
Check this answer by multiplying. We should get the identity matrix if we’ve found the inverse.

27. We can use A-1 to solve a system of equations
To see how, we can re-write a system of equations as matrices.
coefficient matrix
variable matrix
constant matrix

28. left multiply both sides by the inverse of A
This is just the identity
but the identity times a matrix just gives us back the matrix so we have:
This then gives us a formula for finding the variable matrix: Multiply A inverse by the constants.

29. x y find the inverse -2r1+r2 r1-3r2 -r2
This is the answer to the system y

30. Determinants

31. **To find a determinant you must have a SQUARE MATRIX!!**
Determinant - A square array of numbers or variables enclosed between parallel vertical bars.
**To find a determinant you must have a SQUARE MATRIX!!**
Finding a 2 x 2 determinant:

32. Find the determinant:

33. Step 1: Rewrite first two rows of the matrix.
Finding a 3x3 determinant: SARRUS DIAGRAM METHOD
Step 1: Rewrite first two rows of the matrix.

34. 126 - (-52) 126 + 52 = 178 Step 2: multiply diagonals going up!
-224
+10
+162
= -52
Step 2: multiply diagonals going up!
Step 2: multiply diagonals going down!
+12
-126
+240
=126
Step 3: Bottom minus top!
126 - (-52)
= 178

35. 38 - 38 = 0 Step 2: multiply diagonals going up!
-18
+50 +6
= 38
Step 2: multiply diagonals going up!
Step 2: multiply diagonals going down!
- 15 45
+ 8
= 38
Step 3: Bottom minus top!
= 0

36. 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.

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

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

39. Assignment

40. If a matrix has 24 elements , what are the possible order it can have?
Construct a 3x3 matrix whose elements are aij=i+j

41. 3.The bookshop of a school has 10 dozen chemistry books, 8 dozen physics books, and 10 dozen books of maths. Their selling price are Rs.80 ,Rs 60 and Rs 40resp.find the total amount the bookshop will receive from selling all the books? 4.Expand the determinent find x: x x -1 2 x

42. 6. evaluate by sarrus diagram:
1 a b+c
1 b c+a
1 c a+b
7. Prove that
a b c = (b-c)(c-a)(a-b)(a+b+c)
a3 b3 c3

43. Solve the equations by crammer rule: 8. 3x-4y=1 -2x+5y=-3 9
Solve the equations by crammer rule: 8. 3x-4y=1 -2x+5y=-3 9. x+y=1 x+z=-6 x-y-2z= x-y+z=11 X+2y+3z=2 3x+y-z=6

44. test
NOTE:DO ANY THREE

45. 1:
find X and Y ,if 2X+3Y=1
And 5X+7Y=2
2: Construct a 2x2 Matrix A =[a] where a= I2i-3jI/2
3 :If = 2x 4 ; Find the value of x x
4:Evaluate

46. 6.Solve the equation : 5.Let f(x)=x-5x+6 .find f(A) A= 2 0 1 2 1 3
6.Solve the equation :
6x+y-3z=5
x+3y-2z=5
2x+y+4z=8