فصل 1 (المصفوفات)Matrices 1.1 Operations with Matrices (عمليات على المصفوفات) 1.2 Properties of Matrix Operations (خواص عمليات المصفوفات) 1.3 The Inverse of a Matrix (معكوس المصفوفة) 1.4 Elementary Matrices (المصفوفة الأولية) 1.5 Applications of Matrix Operations (تطبيقات على عمليات المصفوفات) 1.1

1.1 Operations with Matrices (عمليات على المصفوفات) Matrix: (i, j)-th entry (or element): number of rows (عدد الصفوف): m number of columns (عدد الأعمدة): n Size (مقاس أو بعد): m×n Square matrix (مصفوفة مربعه): m = n

Equal matrices (المصفوفات المتساوية): two matrices are equal if they have the same size (m × n) and entries corresponding to the same position are equal Ex 1: Equality of matrices (تساوى المصفوفات)

Matrix addition (جمع المصفوفات): Ex 2: Matrix addition (جمع المصفوفات):

Scalar multiplication (الضرب المصفوفه في ثابت قياسي): Matrix subtraction (طرح المصفوفات):: Ex 3: Scalar multiplication and matrix subtraction Find (a) 3A, (b) –B, (c) 3A – B

Sol: (a) (b) (c)

Matrix multiplication (ضرب المصفوفات): should be equal size of C=AB where ※ The entry cij is obtained by calculating the sum of the entry-by-entry product between the ith row of A and the jth column of B

Ex 4: Find AB Sol: Note: (1) BA is not multipliable غير قابلة للضرب (2) Even BA is multipliable قابلة للضرب BAحتى لو أن , AB≠BA

Matrix form of a system of linear equations in n variables الشكل المصفوفى لنظام المعادلات الخطيه: = A x b

Trace operation (أثر المصفوفة) Tr(A): Diagonal matrix (المصفوفه القطريه): a square matrix in which nonzero elements are found only in the principal diagonal ※ It is the usual notation for a diagonal matrix.

1.2 Properties of Matrix Operations (خواص عمليات المصفوفات) Three basic matrix operators, introduced in Sec. 1.1: (1) matrix addition (2) scalar multiplication (3) matrix multiplication Zero matrix (المصفوفه الصفريه): Identity matrix of order n (مصفوفة الوحدة):

Properties of matrix addition and scalar multiplication: then (1) A+B = B+A (Commutative property of addition) (الأبدال) (2) A+(B+C) = (A+B)+C (Associative property of addition) (الدمج) (3) ( cd ) A = c ( dA ) (Associative property of scalar multiplication) (Multiplicative identity property, and 1 is the multiplicative identity for all matrices) (4) 1A = A (5) c( A+B ) = cA + cB (Distributive property of scalar multiplication over matrix addition) (التوزيع عند الضرب فى ثابت) (6) ( c+d ) A = cA + dA (Distributive property of scalar multiplication over real-number addition) حيث c, d أعداد حقيقيه (ثوابت)

Properties of zero matrices (خواص المصفوفه الصفرية): ※ So, 0m×n is also called the additive identity for the set of all m×n matrices ※ Thus , –A is called the additive inverse of A

Properties of matrix multiplication (خواص ضرب المصفوفات): (1) Ａ(BC) = (AB ) C (Associative property of matrix multiplication) (الدمج) (2) Ａ(B+C) = AB + AC (Distributive property of LHS matrix multiplication over matrix addition) (3) (A+B)C = AC + BC (Distributive property of RHS matrix multiplication over matrix addition) (4) c (AB) = (cA) B = A (cB) ※ For real numbers, the properties (2) and (3) are the same. Properties of the identity matrix (خواص مصفوفة الوحدة): : ※ For real numbers, the role of 1 is similar to the identity matrix. However, 1 is unique for real numbers and there could be many identity matrices with different sizes

Ex 3: Matrix Multiplication is Associative (ضرب المصفوفات دامج) Calculate (AB)C and A(BC) for Sol:

Definition of Ak : repeated multiplication of a square matrix: Properties for Ak: (1) AjAk = Aj+k (2) (Aj)k = Ajk where j and k are nonnegative (غير سالبه) integers and A0 is assumed to be I For diagonal matrices فقط للمصفوفة القطرية:

Transpose of a matrix (مدور (منقول) المصفوفة): ※ The transpose operation is to move the entry aij (original at the position (i, j)) to the position (j, i) ※ Note that after performing the transpose operation, AT is with the size n×m

Ex 8: Find the transpose of the following matrix أوجد مدور المصفوفات (b) (c) Sol: (a) (b) (c)

Properties of transposes (خواص على المدور (المنقول)): ※ Properties (2) and (4) can be generalized to the sum or product of multiple matrices. For example, (A+B+C)T = AT+BT+CT and (ABC)T = CTBTAT

Ex 9: Show that (AB)T and BTAT are equal Sol:

Symmetric matrix (مصفوفة متماثلة): A square matrix A is symmetric if A = AT Skew-symmetric matrix (مصفوفة متماثلة تخالفياً): : A square matrix A is skew-symmetric if AT = –A Ex: is symmetric, find a, b, c? Sol:

is a skew-symmetric, find a, b, c? Ex: is a skew-symmetric, find a, b, c? Sol: Note: must be symmetric ※ The matrix A could be with any size, i.e., it is not necessary for A to be a square matrix. ※ In fact, AAT must be a square matrix. Pf:

Real number (الابدال متوفر فى الأعداد الحقيقيه): ab = ba Before finishing this section, two properties will be discussed, which is held for real numbers, but not for matrices: the first is the commutative property of matrix multiplication and the second is the cancellation law (قانون الحذف) Real number (الابدال متوفر فى الأعداد الحقيقيه): ab = ba (Commutative property of real-number multiplication) Matrix (الابدال غير متوفر فى المصفوفات): : Three situations for BA: (Sizes are not the same) (Sizes are the same, but resultant matrices are not equal)

Sow that AB and BA are not equal for the matrices. Ex 4: بين أن Sow that AB and BA are not equal for the matrices. and Sol: (noncommutativity of matrix multiplication)

Notes: (1) A+B = B+A (the commutative law of matrix addition) (2) (the matrix multiplication is not with the commutative law) (so the order of matrix multiplication is very important) (الترتيب فى الضرب مهم جدا)

Real number للأعداد الحقيقية: (Cancellation law for real numbers) Matrix (قانون الحذف فى المصفوفات): (1) If C is invertible (قابله للعكس), then A = B (2) If C is not invertible (غير قابله للعكس), then (Cancellation law is not necessary to be valid) قانون الحذف ليس بالضرورة يكون متوفر

قانون الحذف فى المصفوفات ليس بالضروره يكون متوفر Ex 5: (An example in which cancellation is not valid) Show that AC=BC Sol: So, although , قانون الحذف فى المصفوفات ليس بالضروره يكون متوفر

2.3 The Inverse of a Matrix (معكوس المصفوفة) Inverse matrix: Consider , مصفوفة مربعه then (1) A is invertible (or nonsingular) (2) B is the inverse of A Note: A square matrix that does not have an inverse (ليس لها معكوس) is called noninvertible (or singular) (غير قابله للعكس او شاذه)

معكوس المصفوفه وحيد Theorem نظرية : The inverse of a matrix is unique If B and C are both inverses of the matrix A, then B = C معكوس المصفوفه وحيد. Pf: (associative property of matrix multiplication and the property for the identity matrix) Consequently, the inverse of a matrix is unique. معكوس المصفوفه وحيد Notes: (1) The inverse of A is denoted by معكوس المصفوفة هو