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# M a t r i k s

## by Hartati Yuningsih, Guru Mtk at SMP SMA SMK KENCANA on Nov 26, 2011

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## M a t r i k sPresentation Transcript

•
• 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
• 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 : Arlan Bronto Cery January February March 3 4 1 6 2 3 1 2 4
• 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 M A T R I X
• Matrix is generally notated using capital latter
• 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.
• The first column The second column The third column The column n-th The second row The first row The third row The row n-th
• Example: Matrix A = The first row The second row The first column The second column The third column
• The order matrix A is 2 x 3
• 4 is the second row and the first column
• 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
• a column matrix Is a matrix that only has a column
• A matrix square A square matrix a matrix has the number of row of a matrix equals the number of its column
• Example : rows 4, columns 4 A is matrix the order 4 A = Main diagonal
• A = Upper Triangle Matrix is square matrix which all of the element under the diagonal is zero Upper Triangle Matrix
• B = B is a lower triangle matrix is square matrix which all of the element upper the diagonal is zero Lower Triangle Matrix
• C = Diagonal Matrix is square matrix that all of element is zero, except the element on the diagonal not all of them Diagonal Matrix:
• I = I is matrix Identity that is diagonal matrix that elements at main diagonal value one Pay attention the following matrix
• 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 i th of matrix A to the row of
• ith of new matrix
• b. Change the row of j th 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)
• Transpose matrix A A = IS A t =
• Example :
• 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. 2. Similarity of two matrix
• 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
• and B = A = If Matrix A = Matrix B, so x – 7 = 6  x = 13 2y = -1  y = - ½
• Example 1: Given that K = And L = If K = L, find the value r?
• Answer K = L = p = 6; q = 2p  q = 2.6 = 12 3r = 4q  3r = 4.12 = 48 jadi r = 48 : 3 = 16
• Taking example A = and B = if A t = B, then determine the value x? Example 2:
• Answer : A = = A t = B A t =
• x + y = 1 x – y = 3 2x = 4 so x = 4 : 2 = 2 
• Algebraic Operation on Matrix
• Addition and Subtraction of Matrix
• Scale Multiplication with a Matrix
• Matrix Multiplication with Matrix
• Addition/Subtraction  Two matrix can be summed/reduced if the order of the matrix are same and its statement in one position
• Example 1: and B = A = A + B = + =
• If A = , B = and C = hence(A + C) – (A + B) =…. Example 2:
• (A + C) – (A + B) = A + C – A – B = C – B =  = = Answer
• 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
• Matrix A = Determine matrix represented by 3A 3A = Example :1
• Given Matrix of A = , B = and C = if A – 2B = 3C, So determine a + b ? Example 2 :
• = 3 – = A – 2B = 3C – 2 Answer:
• – = =
• = 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
• 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
• 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
• 26 November 2011 The second row The first row The second column row 1 x column 1 row 1 x column 2 row 2 x column 1 row 2 x column 2 The first column = x … … … …………… row 1 x……. ……… .x column1 A m x n x B n x p = C m x p …………… .. ………… ..
• 26 November 2011 3 4 1 2 7 8 1 x 5 + 2 x 6 1 x 7 + 2 x 8 3 x 5 + 4 x 6 3 x 7 + 4 x 8 5 6 = x Example 1:
• 26 November 2011 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
• 26 November 2011 6 8 5 7 2 4 5 x 1 + 7 x 3 5 x 2 + 7 x 4 6 x 1 + 8 x 3 6 x 2 + 8 x 4 1 3 = x = 26 38 30 44 Example 2:
• 26 November 2011 A = Determine: A x B and B x A and B = Example 3 :
• 26 November 2011 A x B = = = 3 x 5 + (-1) x 8 2 x (-2) + 4 x 1 2 x 5 + 4 x 8 3 x (-2) + (-1) x 1 3 2 4 -1 3 2 4 -1 3 2 4 -1 -2 5 1 8 -7 7 0 42
• 26 November 2011 = 4 (-2) x (-1) + 5 x 4 1 x 3 + 8 x 2 1 x (-1) + 8 x 4 (-2) x 3 + 5 x 2 = 22 19 31 B x A = 3 2 4 -1 -2 5 1 8
• 26 November 2011 conclusion A x B  B x A That is not satisfies the commutative charecteristics
• 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)
• Example 1: Determine the determinant of following matrix!
• Example 2:
• Answer : = = = = = 3 (2x)(-3) – (x – 7)(3) 3 3 -6x – 3x + 21 -18 2 -9x x
• Determinant of a Matrix with Dimension of 3 x 3 + + + - - -
• Determine the determinant of following matrix! Example 2:
• Answer : (-3.2.-7)+(0.13.4)+(5.-5.0)-(5.2.4)- (-3.13.0)+(0.-5.-7) = 42 + 0 + 0 – 40 – 0 – 0 42 - 40 2 = =
• 26 November 2011
• Inverse Matrix
• If A and B are two square matrices with the same dimension such that satisfies AB = BA = I where I is an identity matrix, then
• Matrix of A called inverse from matrix of B given notation of B
• Matrix of B called inverse from matrix of A given notation of A
-1 -1
• 26 November 2011 A = and B = A x B = = -5+6 -3+3 10-10 6-5 = =
• I
Example 1:
• 26 November 2011 A = and B = B x A = = -5+6 -15+15 2-2 6-5 = =
• I
Example 2
• 26 November 2011 Inverse Matrix (2 x 2) if A = Then inverse matrix of A is A -1 =
• 26 November 2011 If determinant of a matrix equals zero then the matrix do not have inverse. This kind of matrix is called singular matrix
• 26 November 2011 Determine the inverse following of a Matrix! A = Example
• 26 November 2011 3 2 -1 -5 Answer
• 26 November 2011 Characteristics inverse of matrix: ( A. B ) -1 = B -1 . A -1 ( A -1 ) -1 = A A .A -1 = A -1 . A = I 1. 2. 3.
• 26 November 2011 Example : Given that A = and B = so (AB) -1 is….
• 26 November 2011 AB = Answer
• 26 November 2011
• Completed the system of linear Equation Using Matrix
• There are two methods can be used to determined the solution of system of linear equation using the matrix approach that are:
• Determinant method
• Inverse matrix method
• Variables
• Using determinant method, the value of x and y found out by the following formula
• Variables b. Using inverse of matrix method, the value of x and y found out by using the following steps.
• Thank You