Chapter 2 … part1

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