Principles of Mathematics - Chapter 1 Matrix PDF

Principles of Mathematics DR. Nerham Youssef  Chapter 1 : Matrices Overall Objectives of Chapter After this lecture you should have a clear of: - What matrices are; - Performing basic operations on matrices; - Special forms of matrices; - The matrix determinant; - How to calculate the inverse...

Principles of Mathematics DR. Nerham Youssef  Chapter 1 : Matrices Overall Objectives of Chapter After this lecture you should have a clear of: - What matrices are; - Performing basic operations on matrices; - Special forms of matrices; - The matrix determinant; - How to calculate the inverse of a 2 by 2 matrix - How to solve the equation by using matrix 2  Vectors and Matrices Matrix is an array of numbers with dimensions M (rows) by N (columns) 3 by 6 matrix  3 0 0 − 2 1 − 2  1 1 3 4 1 −1  −5 2 0 0 0 1    Vector can be considered a 1 x Mmatrix v = (x y z ) Order of matrices… Order 4 by 3: Order 3 by 4: Specifying matrix elements A= aij denotes the element of the matrix A on the ith row and jth column. Types of Matrix For n by n matrices, the identity matrix has 1’s on the diagonal and 0’s everywhere else. 2 by 2 Identity Matrix: 3 by 3 Identity Matrix: If A is n by n, then IA = A and AI = A Operation on Matrices Addition Let A = (aij )mn and B = (bij )mn be matrices. The sum of A and B is the matrix C = (cij )mn denned by cij = aij + bij That is, C is obtained by adding corresponding elements of A And B. We do not dene A + B if A and B do not have the same dimension. Properties of Matrix Addition Let A, B, C, and D be m n matrices. (a) A + B = B + A (so addition is commutative) (b) A + (B + C) = (A + B) + C (so addition is associative) (c) For each m n matrix A, there is a unique m n matrix D such that A+D=O (We write D = - A.) So additive inverses exist. Example (1) Find −5 0 1 1 2 3 3 −2 3 + −2 1 4 5 −6 4 Not valied Operations on Matrices Multiplication 1- multiple by constant: we multiply all value of matrix by the constant 𝑎 𝑏 𝑐 𝑘𝑎 𝑘𝑏 𝑘𝑐 𝑘 × −𝑥 𝑦 = 𝑧 −𝑘𝑥 𝑘𝑦 𝑘𝑧 10 Example (2) 1 2 3 −5 If: 𝐴= 𝐵 = 4 −2 −2 1 4 −2 4 4 Find : A+B, A-B, B-A, 2A, 3B, 2B-A 3 −5 A+B= 1 2 + 4 −2 −2 1 4 −2 4 4 1+4 2 + (−2) 3 + (−5) −2 = = 5 0 −2 + (−2) 1 + 4 4+4 −4 5 8  1 2 3 −5 A-B = − 4 −2 −2 1 4 −2 4 4 1−4 2 − (−2) 3 − (−5) −3 4 8 = = −2 − (−2) 1 − 4 4−4 0 −3 0 −5 1 2 3 B-A = 4 −2 − −2 4 4 −2 1 4 4−1 −2 − 2 −5 − 3 3 −4 −8 = = −2 − (−2) 4 − 1 4−4 0 3 0 12 2A, 3b, 2b-A 2×𝐴 =2× 1 2 3 = 2 4 3 −2 1 4 −4 2 8 3 × 𝐵 = 3 × 4 −2 −5 = 12 −6 −15 −2 4 4 −6 12 12 2 × 𝐵 − 𝐴= 13 Operations on Matrices Multiplication Only possible to multiply of dimensions 𝑚1 ∗ 𝑛1 and 𝑚2 ∗ 𝑛2 if 𝑛1 = 𝑚2 resulting matrix is 𝑚1 ∗ 𝑛2 E.g. Matrix A is 2 × 3 and Matrix B is 3 × 4 resulting matrix is 2 × 4 ❖ Just because A x B is possible doesn’t mean B x A is possible! 14 Example (3) 2 −1 −1 0 2 If A= B= 3 −2 3 1 1 1 −4 Find : 1) AB 2) BA 15 Solution 2 −1 −1 0 2 1) AB= × 3 −2 3 1 1 1 −4 −1 × 2 + 0 × 3 + (2 × 1) ( − 1 × −1) + (0 × −2) + (2 × −4) = 3 × 2 + 1 × 3 + (1 × 1) (3 × −1) + (1 × −2) + (1 × −4) −2 + 0 + 2 1 + 0 + (−8) = 6 + 3 + 1 −3 + (−2) + (−4) 0 −7 = 10 −9 16  2 −1 −1 0 2 2) BA= 3 −2 × 3 1 1 1 −4 2 × −1 + (−1 × 3) 2 × 0 + (−1 × 1) 2 × 2 + (−1 × 1) = 3 × −1 + (−2 × 3) 3 × 0 + (−2 × 1) 3 × 2 + (−2 × 1) 2 × −1 + (−1 × 3) 2 × 0 + (−1 × 1) 2 × 2 + (−1 × 1) −2 + (−3) 0 + (−1) 4 + (−1) = −3 + (−6) 0 + (−2) 6 + (−2) −2 + (−3) 0 + (−1) 4 + (−1) −5 −1 3 = −9 −2 4 −5 −1 3 17 Example (4) i.e. the number of columns in the first matrix must equal the number of rows in the second matrix! 18 Example (3) 1 If A= 4 −2 B= 5 Find : AB, BA SOLUTION 1 AB = 4 −2 × 5 = 4 × 1 + (−2 × 5) = 4 + (−10) = −6 19  1 BA = × 4 −2 5 1 × 4 1 × −2 = 5 × 4 5 × −2 4 −2 = 20 −10 20  Matrix multiplication is NOT commutative In general, if A and B are two matrices then AB≠BA i.e. the order of matrix multiplication is important! EX: 21