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MAT120 Asst. Prof. Dr. Ferhat PAKDAMAR (Civil Engineer) M Blok - M106 pakdamar@gtu.edu.tr Gebze Technical University Department of Architecture Spring 2014/2015 Week 5-6

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Subjects WeekSubjectsMethods 111.02.2015Introduction 218.02.2015 Set Theory and Fuzzy Logic.Term Paper 325.02.2015 Real Numbers, Complex numbers, Coordinate Systems. 404.03.2015 Functions, Linear equations 511.03.2015 Matrices 618.03.2015Matrice operations 725.03.2015MIDTERM EXAM MT 801.04.2015 Limit. Derivatives, Basic derivative rules 908.04.2015 Term Paper presentationsDead line for TP 1015.04.2015 Integration by parts, 1122.04.2015 Area and volume Integrals 1229.04.2015 Introduction to Numeric Analysis 1306.05.2015 Introduction to Statistics. 1413.05.2015Review 15 Review 16 FINAL EXAM FINAL

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2 - 3

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Matrices Definition, and Operations of Matrices: 1 Sums and Scalar Products; 2 Matrix Multiplication Properties of Matrix Operations; Mechanics of Matrix Multiplication The Inverse of a Matrix

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Matrices

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Operations with Matrices Matrix: (i, j)-th entry: row: m column: n size: mn

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i-th row vector j-th column vector row matrix column matrix Square matrix: m = n Operations with Matrices

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Diagonal matrix: Operations with Matrices

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Ex: Operations with Matrices

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Equal matrix: Ex 1 : (Equal matrix) Operations with Matrices

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Matrix addition: Scalar multiplication: Operations with Matrices

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Matrix subtraction: Example: Matrix addition Operations with Matrices

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Matrix multiplication: The i, j entry of the matrix product is the dot product of row i of the left matrix with column j of the right one. Operations with Matrices

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Computation: A x B = C [2 x 2] [2 x 3] Operations with Matrices

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For example: Operations with Matrices

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DETERMINANTS & CRAMERS RULE

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EVALUATE Find the determinant of the matrix: Solution:

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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)]

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EVALUATE Solution:

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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. E W N S Miami (0,0) Bermuda (938,454) Puerto Rico (900,-518)...

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SOLUTION The approximate coordinates of the Bermuda Triangles 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.

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Matrix form of a system of linear equations:

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Partitioned matrices: submatrix Matrix form of a system of linear equations:

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a linear combination of the column vectors of matrix A: linear combination of column vectors of A === Matrix form of a system of linear equations:

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Keywords in Section row vector column vector diagonal matrix trace equality of matrices matrix addition scalar multiplication matrix multiplication partitioned matrix

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Properties of Matrix Operations Three basic matrix operators: (1) matrix addition (2) scalar multiplication (3) matrix multiplication Zero matrix: Identity matrix of order n:

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Then (1) A+B = B + A (2) A + ( B + C ) = ( A + B ) + C (3) ( cd ) A = c ( dA ) (4) 1A = A (5) c( A+B ) = cA + cB (6) ( c+d ) A = cA + dA Properties of matrix addition and scalar multiplication: Commutative law for addition Associative law for addition Associative law for scalar multiplication Unit element for scalar multiplication Distributive law 1 for scalar multiplication Distributive law 2 for scalar multiplication Properties of Matrix Operations

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Notes: (1)0 mn : the additive identity for the set of all mn matrices (2)A: the additive inverse of A Properties of zero matrices: Properties of Matrix Operations

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Properties of the identity matrix: Properties of matrix multiplication: Properties of Matrix Operations

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Transpose of a matrix: Properties of transposes: Properties of Matrix Operations

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A square matrix A is symmetric if A = A T Ex: is symmetric, find a, b, c? A square matrix A is skew-symmetric if A T = A Skew-symmetric matrix: Sol: Symmetric matrix: Properties of Matrix Operations

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is a skew-symmetric, find a, b, c? Note: is symmetric Pf: Sol: Ex: Properties of Matrix Operations

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ab = ba(Commutative law for multiplication) (Sizes are not the same) (Sizes are the same, but matrices are not equal) Real number: Matrix: Three situations of the non-commutativity may occur : Properties of Matrix Operations

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Sol: Example (Case 3): Show that AB and BA are not equal for the matrices. and Properties of Matrix Operations

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(Cancellation is not valid) (Cancellation law) Matrix: (1) If C is invertible (i.e., C -1 exists), then A = B Real number: Properties of Matrix Operations

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Sol: So But Example: An example in which cancellation is not valid Show that AC=BC C is noninvertible, (i.e., row 1 and row 2 are not independent) Properties of Matrix Operations

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Keywords in Section zero matrix identity matrix transpose matrix symmetric matrix skew-symmetric matrix

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The Inverse of a Matrix Note: A matrix that does not have an inverse is called noninvertible (or singular). Consider Then (1) A is invertible (or nonsingular) (2) B is the inverse of A Inverse matrix:

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If B and C are both inverses of the matrix A, then B = C. Pf: the inverse of a matrix is unique. Consequently, the inverse of a matrix is unique. Notes: (1) The inverse of A is denoted by Theorem: Theorem: The inverse of a matrix is unique The Inverse of a Matrix

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detA=a 11 a 22 a 33 +a 21 a 32 a 13 +a 31 a 12 a 23 -a 11 a 32 a 23 -a 31 a 22 a 13 -a 21 a 12 a 33 0

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The Inverse of a Matrix

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Keywords in Section inverse matrix invertible nonsingular singular power

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Example: Find the inverse of the matrix Sol: A Find the inverse of a matrix A by Gauss-Jordan Elimination: The Inverse of a Matrix

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Thus The Inverse of a Matrix

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If A cant be row reduced to I, then A is singular. Note: The Inverse of a Matrix

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Sol: Example: Find the inverse of the following matrix The Inverse of a Matrix

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So the matrix A is invertible, and its inverse is You can check: The Inverse of a Matrix

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Theorem: Theorem: Systems of linear equations with unique solutions If A is an invertible matrix, then the system of linear equations Ax = b has a unique solution given by Pf: ( A is nonsingular) This solution is unique. (Left cancellation property) The Inverse of a Matrix

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Example:

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The Inverse of a Matrix Example:

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The Inverse of a Matrix

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Matrix operations in the Excell

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Have a nice week!