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

Subjects WeekSubjectsMethods Introduction Set Theory and Fuzzy Logic.Term Paper Real Numbers, Complex numbers, Coordinate Systems Functions, Linear equations Matrices Matrice operations MIDTERM EXAM MT Limit. Derivatives, Basic derivative rules Term Paper presentationsDead line for TP Integration by parts, Area and volume Integrals Introduction to Numeric Analysis Introduction to Statistics Review 15 Review 16 FINAL EXAM FINAL

2 - 3

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

Matrices

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

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

Diagonal matrix: Operations with Matrices

Ex: Operations with Matrices

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

Matrix addition: Scalar multiplication: Operations with Matrices

Matrix subtraction: Example: Matrix addition Operations with Matrices

 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

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

For example: Operations with Matrices

DETERMINANTS & CRAMER’S RULE

EVALUATE Find the determinant of the matrix: Solution:

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)]

EVALUATE Solution:

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)...

SOLUTION The approximate coordinates of the Bermuda Triangle’s 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.

Matrix form of a system of linear equations:

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

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:

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

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

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

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

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

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

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

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

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

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

(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

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

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

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:

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

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

The Inverse of a Matrix

Keywords in Section  inverse matrix  invertible  nonsingular  singular  power

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

Thus The Inverse of a Matrix

If A can’t be row reduced to I, then A is singular. Note: The Inverse of a Matrix

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

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

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

Example:

The Inverse of a Matrix Example:

The Inverse of a Matrix

Matrix operations in the Excell

Have a nice week!