Operations on Matrices PDF

Philippine Normal University COLLEGE OF TEACHER DEVELOPMENT Faculty of Science, Technology and The National Center for Teacher Education Mathematics Science, Technology an...

Philippine Normal University COLLEGE OF TEACHER DEVELOPMENT Faculty of Science, Technology and The National Center for Teacher Education Mathematics Science, Technology and Society Center Taft Avenue, Manila UNIT II: Matrix and Matrix Operation  Philippine Normal University COLLEGE OF TEACHER DEVELOPMENT Faculty of Science, Technology and The National Center for Teacher Education Mathematics Science, Technology and Society Center Taft Avenue, Manila Objectives: 1. State Properties of matrix addition, scalar multiplication, and matrix multiplication. 2. Apply properties of matrix operations in simplifying matrices. 3. Describe a square matrix, diagonal matrix, identity matrix, symmetric and skew symmetric. 4. Compute trace of a matrix. Theorem. Commutative Property of Matrix Addition Let A and B be matrices of the same size, m x n. A+B=B+A EXAMPLE: 𝟖 −𝟏𝟐 𝟗 𝟕 W= Y= 𝟒 𝟐 𝟑 −𝟐 W+Y=Y+W Theorem. Associative Property of Matrix Addition Let A, B, and C be matrices of the same size, m x n. A + (B + C) = (A + B ) + C 𝟖 −𝟏𝟐 𝟐 𝟑 𝟗 𝟕 EXAMPLE: W= X= Y= 𝟒 𝟐 −𝟏 −𝟐 𝟑 −𝟐 (W + X) + Y = W + (X + Y) = Theorem. Additive Identity Property of Matrix Addition There is a unique m x n matrix O such that A+ 0 = A for any m x n matrix A. The matrix O is called the m x n additive identity or zero matrix. 𝟖 −𝟏𝟐 𝟎 𝟎 𝟖 −𝟏𝟐 + = 𝟒 𝟐 𝟎 𝟎 𝟒 𝟐 Theorem. Additive Inverse Property of Matrix Addition To each m x n matrix A, there is a unique m x n matrix D such that A+D=0 We shall write D as (- A), so that A + D = 0 can be written as A + (-A) = 0 The matrix (-A) is called the additive inverse or negative of A. 𝟖 −𝟏𝟐 −𝟖 𝟏𝟐 + −𝟒 −𝟐 = 𝟎 𝟎 𝟒 𝟐 𝟎 𝟎 Theorem. Associative Property of Scalar Multiplication Let A be a matrix and r and s are scalars. r(sA) = (rs)A Example: 𝟖 −𝟏𝟐 Let A = and r = 2 and s = -1 𝟒 𝟐 r(s A) = rs (A) = Theorem. Distributivity I Property of Scalar Multiplication Let A be a matrix and r and s be scalars. (r + s)A = rA + sA 𝟖 −𝟏𝟐 Let A = and r = 3 and s = -4 𝟒 𝟐 𝟖 −𝟏𝟐 (r + s) (A) = (3 + -4) 𝟒 𝟐 𝟖 −𝟏𝟐 𝟖 −𝟏𝟐 rA + sA = (3) + (-4) 𝟒 𝟐 𝟒 𝟐 Theorem. (Distributivity II) Property of Scalar Multiplication Let A and B be matrices of appropriate sizes, and let r and s be scalars. r(A + B) = rA + rB 1 −3 0 3 A= 6 9 B = −1 0 𝑟=6 5 15 4 −12 1 −3 0 3 r(A + B) = (6) 6 9 + −1 0 5 15 4 −12 1 −3 0 3 rA + rB) = (6) 6 9 + (6) −1 0 5 15 4 −12 Theorem Property of Scalar Multiplication Let A and B be matrices of appropriate sizes, and let r and s be scalars. A(rB) = r(AB) = (rA)B 1 −3 A= 6 9 2 3 −1 B= 𝑟=2 5 15 2 5 1 1 −3 2 3 −1 A(rB) = 6 9 (2) = 2 5 1 5 15 1 −3 2 3 −1 r(AB) = 2 6 9 2 5 1 5 15 1 −3 (rA)B = (2) 6 9 2 3 −1 = 5 15 2 5 1 Theorem Associative Property of Matrix Multiplication Let A, B, and C be matrices of appropriate sizes. A(BC) = (AB)C 1 −3 A= 6 −1 C= 4 2 3 B= −2 2 3 1 −3 A(BC) = 6 −1 3 4 2 = 3 −2 2 1 −3 −1 (AB)C = 6 3 4 2 = 3 −2 2 Theorem. Right Distributive Property of Matrix Multiplication Let A,B, and C be matrices of appropriate sizes. A(B + C) = AB + AC 1 −3 A= 6 −1 −2 3 B= C= −2 2 3 1 1 −3 A(B + C) = 6 −1 −2 3 + = 3 1 −2 2 1 −3 1 −3 AB + AC = −1 −2 6 3 + 6 3 = 3 1 −2 2 −2 2 Theorem. Left Distributivity Property of Matrix Multiplication Let A,B, and C be matrices of appropriate sizes. (A + B)C = AC+ BC −1 −2 C = 𝟑 𝟒 −𝟏 A= B= 3 1 −𝟏 −𝟐 (A + B) C = + 𝟑 𝟒 −𝟏 𝟑 𝟏 −𝟏 −𝟐 AC + BC = 𝟑 𝟒 −𝟏 + 𝟏 𝟑 𝟒 −𝟏 𝟑 REMARKS 1. AB may not be equal to BA 2. AB = 0 does not imply that A = 0 or B = 0 3. AB = AC does not imply that B = C. They may be different. In each item, compute for AB and BA. −𝟐 𝟏 𝟒 𝟎 𝟎 𝟑 B = −𝟏 𝟐 𝟏. 𝑨 = 𝟑 𝟑 𝟏 −𝟏 2. 𝑨 = B = 𝟒 𝟒 −𝟏 𝟏 Find AC and BC. 𝟏 𝟑 𝟐 𝟒 𝟏 −𝟐 3. 𝑨 = B= C= 𝟎 𝟏 𝟐 𝟑 −𝟏 𝟐 Definition. Identity Matrix Identity Matrix. A square matrix (n x n matrix) where the elements on its leading diagonal (the diagonal running from top left to bottom right) are 1 and the rest are of value 0. 𝟏 𝟎 𝟎 𝟏𝟎 Examples: 𝟎 𝟏 𝟎 𝟎𝟏 𝟎 𝟎 𝟏 Properties of Identity Matrix If A is a matrix of size m x n, then the properties below are true. 1. AIn = A 2. ImA = A Example: Multiplication by an Identity Matrix 3 2 3 2 1 0 0 −4 −4 1 0 a. −2 5 = −2 5 b. 0 1 0 2 = 2 0 1 6 7 6 7 0 0 1 0 0 Name the identity matrix that should be multiplied to the given matrix to get the same matrix. 1 0 0 1 5 12 0________ 1 0 1 5 12 = −3 0 4 0 0 1 −3 0 4 Definition. Powers of Matrices If p is a positive integer and A is a square (n x n) matrix, we define A p = AA … A and A0 = I n p factors −1 2 −1 2 70 Example : = 3 1 3 1 07 2 3 2 1 1 If A = 1 1 and B = 1 4 0 , determine if 𝐴3 and 𝐵5 are defined 0 −3 3 −1 1 a. 𝐴3 b. 𝐵5 Definition. Diagonal Matrix Definition. Diagonal Matrix. It is a square matrix whose entries above and below the main diagonal are 0. 𝟐 𝟎 𝟎 𝟕𝟎 Example : 𝟎 𝟑 𝟎 𝟎𝟕 𝟎 𝟎 𝟏 𝟐 𝟎 𝟎 If B = 𝟎 𝟒 𝟎 , what is 𝑩𝟐 ? 𝒂𝒏𝒅 𝑩𝟑 ? 𝟎 𝟎 𝟑 Definition. Transpose of a Matrix The matrix obtained from any given matrix A, by interchanging the rows and columns, written as AT (to be read as A – Transpose) 𝟐 𝟑 −𝟐 𝟐 𝟏 𝟎 Examples : A= 𝟏 𝟕 𝟔 𝐀𝐓 = 𝟑 𝟕 𝟒 𝟎 𝟒 𝟏 −𝟐 𝟔 𝟏 −𝟏 𝟐 −𝟏 𝟑 𝟐 A= 𝟑 𝟒 𝐀𝐓 = 𝟐 𝟓 𝟐 𝟒 𝟓 𝟐 𝟎 𝟎 𝟐 𝟎 𝟎 B= 𝟎 𝟒 𝟎 𝐁𝐓 = 𝟎 𝟒 𝟎 𝟎 𝟎 𝟑 𝟎 𝟎 𝟑 Definition. Transpose of a Matrix Theorem If r is a scalar and A and B are matrices, then (A T ) T = A Proof: (1) Let A be an m x n matrix. Observe that AT has size n x m and ( AT)T has size m x n, the same as A. To show that (AT)T = A, we must show that the ijth entries are the same as A. Let aij be the ijth entry of A. Then aij is the jith entry of AT, and the ijth entry of (AT)T. This proves property 1. Definition. Transpose of a Matrix Theorem If r is a scalar and A and B are matrices, then 1. (A + B) T = A T + B T 2. (AB) T = B T A T 3. (rA) T = rAT 𝟎 𝟐 −𝟏 𝟏 𝟑 −𝟏 1. (A + B) T = A T + B T A = −𝟐 𝟎 −𝟒 B = −𝟐 𝟖 𝟒 𝟏 𝟒 𝟎 𝟏 𝟔 𝟓 (A + B) T = A T+ B T Definition. Transpose of a Matrix Theorem If r is a scalar and A and B are matrices, then 1. (A + B) T = A T + B T 2. (AB) T = B T A T 3. (rA) T = rAT 𝟐 2. (AB) T = B T A T Example: A= 𝟏 B= 𝟒 𝟓 𝟑 (AB) T ATBT BTAT Definition. Transpose of a Matrix Theorem If r is a scalar and A and B are matrices, then 1. (A + B) T = A T + B T 2. (AB) T = B T A T 3. (rA) T = rAT 3. (rA) T = rAT 𝟎 𝟐 −𝟏 A = −𝟐 𝟎 −𝟒 r=-3 𝟏 𝟒 𝟎 𝑻 𝟎 𝟐 −𝟏 (rA) T = (−𝟑) −𝟐 𝟎 −𝟒 𝟏 𝟒 𝟎 𝑻 𝟎 𝟐 −𝟏 rAT = (−3) −𝟐 𝟎 −𝟒 𝟏 𝟒 𝟎 Definition. Symmetric A matrix A is called symmetric if, AT = A. That is, A is symmetric if it is a square matrix for which aij = aji. Example: 1 2 3  1 2 3  A =  2 4 5 AT = 2 4 5     3 5 6 3 5 6 Remark A matrix is symmetric if its entries are symmetric with respect to the main diagonal. Definition. Skew Symmetric A matrix is called skew symmetric if AT = - A. Example: 𝟎 −𝟐 𝟏 𝑨𝑻 = 𝟐 𝟎 𝟒 𝟎 𝟐 −𝟏 −𝟏 −𝟒 𝟎 A = −𝟐 𝟎 −𝟒 𝟏 𝟒 𝟎 𝟎 −𝟐 𝟏 -A = 𝟐 𝟎 𝟒 −𝟏 −𝟒 𝟎 Remark : If A is a skew symmetric matrix then the elements of the main diagonal aij= 0. Definition. Trace of a Matrix If A = [ aij] is an n x n matrix, then the trace of A, Tr(A), is defined as the sum of all elements on the main diagonal of A, Tr(A) = a11 + a22 + …+ ann 1 −1 3 Example: Compute tr 5 0 2 = 1+0+4=5 2 1 4 1 0 0 5 Example: Compute tr 0 3 2 4 = 1 + 3 + (-4) + 8 = 8 3 1 −4 2 2 −2 3 8  EXAMPLES Consider the matrices. 3 0 1/2 0 0 1 0 0 4 −1 1 4 2 A = −1 2 B= C= D= 0 0 0 E= 0 1 0 0 2 3 1 5 1 1 0 0 4 0 0 1 Compute the following (where possible). 1. A(BC) 2. 𝑩𝑻 + 𝟓𝑪𝑻 3. Tr (4𝑬𝑻 − 𝑫) END for this Session ☺