Chapter 2 … part1 Matrices Linear Algebra S 1

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

Ch2_3 2.1 Addition, Scalar Multiplication, and Multiplication of Matrices a ij : 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 column is called a ……….. matrix. For a square n  n matrix A, the main diagonal is: ………………. We can denote the matrix by …………….. Note:

Ch2_4 Definition Two matrices are equal if: 1) ………………….. 2) …………………….. Example 1

Ch2_5 Addition of Matrices Definition If A and B be matrices of the …………….. then the sum A + 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: ……………………

Ch2_6 Example 2 Determine A + B, 3A, A + C, A-B Solution

Ch2_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 be matrices, 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_8 Example 3 Compute the linear combination: for: Solution

Ch2_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 ……………..

Ch2_10

Ch2_11 Note: Example 4 Let C = AB, Determine c 23.

Ch2_12 Example 5 Solution Note. In general, ……………

Ch2_13 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 ……. Special Matrices

Ch2_14 Theorem 2.1 Let A be m  n matrix and O mn be the zero m  n matrix. Let B be an n  n square matrix. O n and I n be the zero and identity n  n matrices. Then: 1) A + O mn = O mn + A = ……. 2) BO n = O n B = ……… 3) BI n = I n B = ……… Example 6

Ch2_15 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. AI n = I n A =……… (where I n is the identity matrix) 5. k(AB) = ………= ……… Note: AB  BA in general.Multiplication of matrices is not commutative. Theorem Algebraic Properties of Matrix Operations

Ch2_16 Example 7 Compute ABC. Solution

Ch2_17 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. Note : Example 8

Ch2_18 Powers of Matrices Theorem 2.3 If A is an n  n square matrix and r and s are nonnegative integers, then 1. A r A s = ………. 2. (A r ) s = ………. 3. A 0 = ……… (by definition) Definition If A is a square matrix and k is a positive integer, then

Ch2_19 Example 9 Solution Example 10 Simplify the following matrix expression. Solution

Ch2_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 A p =0 is called the ……………………. of the matrix. Example 11

Ch2_ Symmetric Matrices Definition The …………….. of a matrix A, denoted ………, is the matrix whose ………….. are the ………. of the given matrix A. Example 12 Determine the transpose of the following matrices:

Ch2_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. (A t ) t = ………...

Ch2_23 Symmetric Matrix match Definition Let A be a square matrix: 1) If ………... then A called ………………... matrix. 2) If ………... then A called ………………... matrix. Example 13 symmetric matrices

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

Ch2_25 Example 15 Proof Let A and B be symmetric matrices of the same size. Prove that the product AB is symmetric if and only if AB = BA.

Ch2_26 Example 16 Proof Let A be a symmetric matrix. Prove that A 2 is symmetric.

Ch2_27 Definition Let A be a square matrix. The ………… of A denoted by …….. is the …………………………………. of A. 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.tr(A + B) = ………………….. 2.tr(kA) = …………. 3.tr(AB) = ………… 4.tr(A t ) = ………….. Theorem : Properties of Trace.