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Understanding, N o tation, and Dimension of Matrix Types of Matrix

Understanding, N o tation, and Dimension of Matrix Types of Matrix Transpose and Similarity of a matrix Operation of the Matrix Inverse and Determinant of the Matrix Completed the system of Linear Using Matrix. Understanding, Notation, and Demension Of Matrix. Understanding Of Matrix

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Understanding, N o tation, and Dimension of Matrix Types of Matrix

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  1. Understanding, Notation, and Dimension of Matrix Types of Matrix Transpose and Similarity of a matrix Operation of the Matrix Inverse and Determinant of the Matrix Completed the system of Linear Using Matrix

  2. Understanding, Notation, and Demension Of Matrix • Understanding Of Matrix • Pay attention the following illustration • Mr. Andi note is student absent in last three year, that is January, February and March to 3 student that is Arlan, Bronto, and cery like at the table. On the table can be written :

  3. M A T R I X is a rectangular array of numbers, consists of rows and columns and is written using brackets or parentheses. The entries of a matrix are called elements of matrix . An element of a matrix is addressed by listing the row number and then column number

  4. Matrix is generally notated using capital latter

  5. 2. The order of the matrix A matrix of A has m rows and n column is called as matrix of dimension on order m x n, and so notated of “A(mxn)”. To more understand the definition of the element of a matrix.

  6. The first row The second row The third row The row n-th The column n-th The third column The second column The first column

  7. Example: The first row Matrix A = The second row The first column The second column The third column • 4is the second row and the first column • The order matrix A is 2 x 3

  8. a row matrix Is a matrix that only has a row A = ( 1 3 5), and B = ( -1 0 4 7) The order matrix is and

  9. a column matrix Is a matrix that only has a column

  10. A matrix square A square matrix a matrix has the number of row of a matrix equals the number of its column

  11. Example : rows 4, columns 4 A is matrix the order 4 A = Main diagonal

  12. Upper Triangle Matrix A = Upper Triangle Matrix is square matrix which all of the element under the diagonal is zero

  13. Lower Triangle Matrix B = B is a lower triangle matrix is square matrix which all of the element upper the diagonal is zero

  14. Diagonal Matrix: C = Diagonal Matrix is square matrix that all of element is zero, except the element on the diagonal not all of them

  15. Pay attention the following matrix I = I is matrix Identity that is diagonal matrix that elements at main diagonal value one

  16. Transpose and Similarity of a Matrix • Transpose of a Matrix • Let A is a matrix whit dimension of (m x n). From the matrix of A we can formed a new matrix that obtained by following method: • a. Change the line of ith of matrix A to the row of • ith of new matrix • b. Change the row of jth of matrix A to the line of • jth of new matrix • The new matrix that resulted is called transpose from matrix of A symbolized with A’ or From the above changess, the dimension of A’ is (n x m)

  17. A = Transpose matrix A IS At =

  18. Example :

  19. 2. Similarity of two matrix let A = (aij) ang B = (bij) are two matrices with the same dimension. Matrix of A is callled equal with matrix of B id the element that located on the two matrices has the same value.

  20. One located element with the same value One located element with the same value One located element with the same value One located element with the same value

  21. A = and B = If Matrix A = Matrix B, so x – 7 = 6  x = 13 2y = -1  y = -½

  22. Example 1: Given that K = And L = If K = L, find the value r?

  23. Answer K = L = p = 6; q = 2p  q = 2.6 = 12 3r = 4q  3r = 4.12 = 48 jadi r = 48 : 3 = 16

  24. Example 2: Taking example A = and B = if At = B, then determine the value x?

  25. Answer : A = At = At = B =

  26. x + y = 1 x – y = 3 2x = 4 so x = 4 : 2 = 2 

  27. Algebraic Operation on Matrix • Addition and Subtraction of Matrix • Scale Multiplication with a Matrix • Matrix Multiplication with Matrix

  28. Addition/Subtraction Two matrix can be summed/reduced if the order of the matrix are same and its statement in one position

  29. Example 1: A = and B = A + B = + =

  30. Example 2: If A = , B = and C = hence(A + C) – (A + B) =….

  31. Answer (A + C) – (A + B) = A + C – A – B C – B =  = = =

  32. Scale Multiplication With a Matrix Let k Є R and A is a matrix with dimension of m x n . Multiplication of real number k by matrix of A is a new matrix which is also has dimension of m x n that obtained by multiplying each element A by real number of k and notates kA

  33. Example :1 Matrix A = Determine matrix represented by 3A 3A =

  34. Example 2 : , B = Given Matrix of A = and C = if A – 2B = 3C, So determine a + b ?

  35. Answer: A – 2B = 3C – 2 = 3 – =

  36. – = =

  37. = a – 2 = -3  a = -1 4 – 2a – 2b = 6 4 + 2 – 2b = 6 6 – 2b = 6 -2b = 0  b = 0 Become a + b = -1 + 0 = -1

  38. Matrix Multiplication with Matrix The Product Of Two Matrices A and B can be got when satisfies the relation A m x n = B p x q = AB m x q Equal

  39. The number of column of matrix A should equal the number of rows of matrix B, the product, that is AB has order of m x q. when m is the number of rows of matrix A and q is the number of column of matrix B

  40. Am x n x Bn x p = Cm x p The first column The second column …………… The first row x The second row … … … row 1 x column 1 row 1 x column 2 row 1 x……. = row 2 x column 1 row 2 x column 2 ………….. …………….. ……….x column1 41 02 Oktober 2014

  41. Example 1: 1 2 5 6 7 8 x 3 4 1 x 5 + 2 x 6 1 x 7 + 2 x 8 = 3 x 5 + 4 x 6 3 x 7 + 4 x 8 42 02 Oktober 2014

  42. 1 x 5 + 2 x 6 1 x 7 + 2 x 8 = 3 x 5 + 4 x 6 3 x 7 + 4 x 8 17 23 = 39 53 43 02 Oktober 2014

  43. Example 2: 5 7 1 3 2 4 x 6 8 5 x 1 + 7 x 3 5 x 2 + 7 x 4 = 6 x 1 + 8 x 3 6 x 2 + 8 x 4 26 38 = 30 44 44 02 Oktober 2014

  44. Example 3 : A = and B = Determine: A x B and B x A 45 02 Oktober 2014

  45. -1 -1 -1 3 3 3 -2 5 2 2 2 4 4 4 1 8 -7 7 0 42 A x B = 3 x (-2) + (-1) x 1 3 x 5 + (-1) x 8 = 2 x (-2) + 4 x 1 2 x 5 + 4 x 8 = 46 02 Oktober 2014

  46. -2 5 -1 3 B x A = 1 8 2 4 (-2) x 3 + 5 x 2 (-2) x (-1) + 5 x 4 = 1 x 3 + 8 x 2 1 x (-1) + 8 x 4 4 22 = 19 31 47 02 Oktober 2014

  47. conclusion A x B  B x A That is not satisfies the commutative charecteristics 48 02 Oktober 2014

  48. Determinant of a Matrix Determinant of a Matrix with Dimension of 2 x 2 Determinant from a matrix of A notated with det (A), , or is a certain value with the size is equal (ad – bc)

  49. Example 1: Determine the determinant of following matrix!

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