Matrices

1. Matrices

2. 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.

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

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

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

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-1does 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. 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. -2r1+r2 r1-3r2 -r2 find the inverse x This is the answer to the system y

30. Determinants

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

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

35. -18 +50 +6 = 38 45 - 15 + 8 Step 2: multiply diagonals going up! Step 2: multiply diagonals going down! = 38 Step 3: Bottom minus top! 38 - 38 = 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 2 -4 3 3 1 2 7 6 1 5. find x: x 2 -1 2 5 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 1 1 1 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. x+y=1 x+z=-6 x-y-2z=3 10. 2x-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 2 4 = 2x 4 ; Find the value of x 5 1 6 x 4:Evaluate 1 3 5 2 6 10 31 11 38

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