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Fundamentals of Matrices: Addition, Scalar Multiplication, and Matrix Multiplication

This chapter covers fundamental operations on matrices including addition, scalar multiplication, and matrix multiplication. It defines matrices as rectangular arrays of numbers and explains their properties, including equality and special types such as zero and identity matrices. The chapter also illustrates how these operations follow specific algebraic properties, emphasizes the importance of matrix size compatibility, and discusses notable matrix characteristics like transpose, symmetric, idempotent, and nilpotent matrices. This foundational knowledge is crucial for advanced matrix theory and linear algebra.

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Fundamentals of Matrices: Addition, Scalar Multiplication, and Matrix Multiplication

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  1. Linear Algebra Chapter 2 … part1 Matrices S 1

  2. 2.1 Addition, Scalar Multiplication, and Multiplication of Matrices Definition A matrixis a rectangular array of numbers. The numbers in the array are called the elementsof the matrix. Denoted by: A,B,…capital letter. Ch2_2

  3. 2.1 Addition, Scalar Multiplication, and Multiplication of Matrices Note: • aij: the element of matrix A in row….. and column …… • we say it is in the ……………….. • The size of a matrix = number of …... × number of ….….. = ……. • If n=m the matrix is said to be a …….. matrix with size = …... or …. • A matrix that has one row is called a …… matrix. • A matrix that has one columnis called a ………..matrix. • For a squarenn matrix A, the main diagonal is: ………………. • We can denote the matrix by …………….. Ch2_3

  4. Example 1 Definition Two matrices are equal if: 1) ………………….. 2) ……………………..

  5. Addition of Matrices • Definition • If A and B be matrices of the …………….. then the • sumA + B=C will be of the ……….. size and • …………………… • If • Let A be a matrix and k be a scalar. The scalar multiple of A by k , denoted …………will be the same size as A. • …………………… • The matrix (-1)A= -A called the …………… of A. • Let A and B of the same size then: A - B= A +(-B)=C and: • ……………………

  6. Determine A + B , 3A , A + C , A-B Example 2 Solution

  7. Definition • A……. matrix all of it’s elements are zero. If the zero matrix is of a square size n×n it will be denoted by . Theorem2.2: Let A,B,C bematrices, be scalars. Assume that the size of the matrices are such that the operations can be performed, let 0 be the zero matrix. Properties of matrix addition and scalar multiplication: Ch2_7

  8. Example 3 Compute the linear combination: for: Solution Ch2_8

  9. Multiplication of Matrices Definition 1) If the number of ……….. in A = the number of …….. in B. The product AB then exists. Let A: …….. matrix, B: ………. matrix, The product matrix C=AB is a ………. matrix. 2) If the number of ……….in A the number of ……..in B then The product AB ……………..

  10. Ch2_10

  11. Note: Example 4 Determine c23. Let C = AB, Ch2_11

  12. Example 5 Solution Note.In general, ……………

  13. Special Matrices Definition 1) A …….. matrix is a matrix in which all the elements are zeros. 2) A ……….. matrix is a square matrix in which all the elements ……………………………………... 3) An ……….. matrix is a diagonal matrix in which every element in the main diagonal is …….

  14. Example 6 Theorem 2.1 Let A be m n matrix and Omn be the zero m n matrix. Let B be an n n square matrix. On and In be the zero and identity n n matrices. Then: 1) A + Omn = Omn + A = ……. 2) BOn = OnB = ……… 3) BIn = InB = ………

  15. 2.2 Algebraic Properties of Matrix Operations Theorem 2.2 -2 Let A, B, and C be matrices and k be a scalar. Assume that the size of the matrices are such that the operations can be performed. Properties of Matrix Multiplication 1. A(BC) = …………. Associative property of multiplication 2. A(B + C) = ………… Distributive property of multiplication 3. (A + B)C = ………… Distributive property of multiplication 4. AIn = InA =……… (whereIn is the identity matrix) 5. k(AB) = ………= ……… Note: AB BA in general. Multiplication of matrices is not commutative.

  16. Compute ABC. Example 7 Solution

  17. Note: • In algebra we know that the following cancellation laws apply. • If ab = ac and a  0 then ……….. • If pq = 0 then ……….. or ………. • However the corresponding results are not true for matrices. • AB = AC………………. that B = C. • PQ = O………………… that P = O or Q = O. Example 8

  18. Powers of Matrices Definition If A is asquare matrix and k is a positive integer, then Theorem 2.3 If A is an n n square matrix and r and s are nonnegative integers, then 1. ArAs = ………. 2. (Ar)s = ………. 3. A0 = ……… (by definition)

  19. Example 10 Simplify the following matrix expression. Solution Example 9 Solution

  20. Idempotent and Nilpotent Matrices Definition • A square matrix A is said to be: • ……………….if ………….. • ………………. if there is a positive integer p s.t ……….… The least integer p such that Ap=0 is called the • ……………………. of the matrix. Example 11

  21. Example 12 2.3 Symmetric Matrices Definition The…………….. of a matrix A, denoted ………, is the matrix whose ………….. are the ………. of the given matrix A. Determine thetranspose of the following matrices:

  22. Theorem :Properties of Transpose Let A and B be matrices and k be a scalar. Assume that the sizes of the matrices are such that the operations can be performed. 1. (A + B)t = ………... Transpose of a sum 2. (kA)t = ...… Transpose of a scalar multiple 3. (AB)t = ………... Transpose of a product 4. (At)t = ………...

  23. match match Symmetric Matrix Definition Let A be a square matrix: 1) If ………...then A called………………... matrix. 2) If ………... then A called………………...matrix. Example 13 symmetric matrices

  24. Example 14 Let A and B be symmetric matrices of the same size. C = aA+bB, a,b are scalars. Prove that C is symmetric. Proof Ch2_24

  25. Example 15 Let A and B be symmetric matrices of the same size. Prove that the product AB is symmetric if and only if AB = BA. Proof

  26. Example 16 Let A be a symmetric matrix. Prove that A2 is symmetric. Proof

  27. Definition Let A be a square matrix. The ………… of A denoted by ……..is the …………………………………. of A. Theorem :Properties of Trace . • Let A and B be matrices and k be a scalar. Assume that the sizes of the matrices are such that the operations can be performed. • tr(A + B) = ………………….. • tr(kA) = …………. • tr(AB) = ………… • tr(At) = ………….. Ch2_27

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