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MAT 2401 Linear Algebra 2.1 Operations with Matrices http://myhome.spu.edu/lauw HW... If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE. GJE is the corner stone of this class, you really need to figure it out. Today Written HW Again, today may be longer. It is more efficient to bundle together some materials from 2.2. Next class session will be shorter. Preview Look at the algebraic operations of matrices “term-by-term” operations • Matrix Addition and Subtraction • Scalar Multiplication Non-“term-by-term” operations • Matrix Multiplication Matrix If a matrix has m rows and n columns, then the size (dimension) of the matrix is said to be mxn. 1 1 2 m 2 n Notations Matrix A aij j th i th aij Notations Matrix Example: A aij 1 1 1 4 A 2 2 5 11 4 6 8 24 j th i th aij a11 a23 Special Cases Row Vector b1 b2 bn Column Vector c1 c 2 cm Matrix Addition and Subtraction Let A = [aij] and B = [bij] be mxn matrices Sum: A + B = [aij+bij] Difference: A-B = [aij-bij] (Term-by term operations) Example 1 1 2 A 3 1 0 2 B 3 2 A B A B Scalar Multiplication Let A = [aij] be a mxn matrix and c a scalar. Scalar Product: cA=[caij] Example 2 1 2 A 3 1 2A Matrix Multiplication Define multiplications between 2 matrices Not “term-by-term” operations Motivation The LHS of the linear equation consists of two pieces of information: • coefficients: 2, -3, and 4 • variables: x, y, and z 2x 3y 4z 5 Motivation Since both the coefficients and variables can be represented by vectors with the same “length”, it make sense to consider the LHS as a “product” of the corresponding vectors. 2x 3y 4z 5 x 2 3 4 y 5 z Row-Column Product a1 a2 b1 b an 2 a1b1 a2b2 bn same no. of elements anbn Example 3 2 2 1 3 2 4 1 2 Matrix Multiplication a11 ai1 am1 a12 ai 2 am 2 a1 p b11 b aip 21 b amp p1 ai1b1 j ai 2b2 j b1 j b2 j bpj aip bpj b1n b2 n bpn j th i th cij Example 4 1 2 0 1 1 0 1 0 Example 5 (a) 4 2 1 2 1 0 1 2 3 1 2 1 Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix? Example 5 (b) 1 2 3 2 2 3 1 3 Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix? Example 5 (c) 1 1 1 2 Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix? Example 5 (d) 1 1 2 1 Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix? 1 Remark: A 1 2 , B 1 Example 5 (e) 1 1 1 1 1 1 1 1 Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix? 1 1 1 1 , B Remark: A 1 1 1 1 Example 5 (f) 1 0 1 2 0 1 3 4 Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix? 1 0 1 2 I , A Remark: 0 1 3 4 Interesting Facts The product of mxp and pxn matrices is a mxn matrix. In general, AB and BA are not the same even if both products are defined. AB=0 does not necessary imply A=0 or B=0. Square matrix with 1 in the diagonal and 0 elsewhere behaves like multiplicative identity. Identity Matrix nxn Square Matrix 1 0 I In 0 0 1 0 0 0 1 Zero Matrix mxn Matrix with all zero entries 0 0mn 0 0 0 0 0 0 0 0 0 Representation of Linear System by Matrix Multiplication x yz 4 2 x 2 y 5 z 11 4 x 6 y 8 z 24 Representation of Linear System by Matrix Multiplication x yz 4 2 x 2 y 5 z 11 4 x 6 y 8 z 24 Representation of Linear System by Matrix Multiplication x y z 4 2 x 2 y 5 z 11 4 x 6 y 8 z 24 Representation of Linear System by Matrix Multiplication Let x yz 4 2 x 2 y 5 z 11 4 x 6 y 8 z 24 1 1 1 x 4 A 2 2 5 , X y , b 11 4 6 8 z 24 Then the linear system is given by Remark It would be nice if “division” can be defined such that: Let 1 1 1 x 4 A 2 2 5 , X y , b 11 4 6 8 z 24 Then the linear system is given by (2.3) Inverse HW... If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE. GJE is the corner stone of this class, you really need to figure it out.