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CSUN ME 501A - Matrix Multiplication, Inverse Matrices, and Determinants

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1Matrix Multiplication, Inverse Matrix Multiplication, Inverse Matrices, and DeterminantsMatrices, and DeterminantsLarry CarettoMechanical Engineering 501ASeminar in Engineering Seminar in Engineering AnalysisAnalysisAugust 26, 20042Overview• Review last lecture• Background for matrix multiplication• General rule and examples of matrix multiplication• Definition, computation and use of determinants• Use of determinants in computing matrix inverses• Example problems3Review last lecture• Vector has a magnitude and a direction– Can represent by componentsθf1f2fxy• f·dx = |f||dx|cos(θ)• f = f1i + f2j+ f3k or f = [f1 f2 f3] • Unit vectors i = [1 0 0], j = [0 1 0], k = [0 0 1] in x, y, z directions• f·dx = f1dx1+ f2dx2+ f3dx34Review Matrix Basics• Array of numbers with n rows and m columns• Components are a(row)(column)⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡=nmnnnmmmaaaaaaaaaaaaaaaaLLMOMMMMOMMMLLLLLL321333323122322211131211A• Size of matrix (n x m) is number of rows and columns• Square matrix: m = n5Review Diagonal Matrix• The diagonal matrix A is a square matrix with nonzero components only on the principal diagonal⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡=naaaaLLMOMMMMOMMMLLLLLL000000000000321A• Components of A are aiδij, where δijis the Kroenecker delta⎩⎨⎧≠==jijiij01δ6Review Matrix Operations• Can add or subtract matrices if they are the same size– C = A ± B only valid if A, B, and C have the same size (rows and columns)– Components of C, cij= aij± bij• Multiplication by a scalar: C = xA– C and A have the same size (rows and columns)– Components of C, cij= xaij27Review Null (0)/Unit (I) Matrices• For any matrix, A, A + 0 = 0 + A = A;IA = AI = A and 0 A = A 0 = 0• The unit (or identity) matrix is a square matrix; the null matrix need not be square⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎢⎣⎡=10000100001000010000000000000000LLMOMMMMOMMMLLLLLLLLMOMMMMOMMMLLLLI08Review Matrix Transpose• Transpose of A denoted as AT(or A’)• Reverse rows and columns; for B = AT–bij= aji–If A is (n x m), B = ATis (m by n)⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=⎥⎦⎤⎢⎣⎡−−=0621214302146123TAA9Review Row/Column Vectors• Matrices with only one row or only one column are called row or column vectors (or matrices)[][]⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡=⎥⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎢⎣⎡===nnmmccccccccrrrrrrrrMMMMLLLL32113121113211131211cr10Matrix Multiplication• Not an intuitive operation. Look at two coordinate transformations as example 2221212212111122212122121111ybybzybybzxaxayxaxay+=+=+=+=][][][][2221212221211121222212112212111111xaxabxaxabzxaxabxaxabz+++=+++=• Substitute equations for y in terms of x into equations for z1and z211Matrix Multiplication II• Rearrange last set of equations to get direct transformation from x to z22212122222122112122112122121112221212111211211111][][][][xcxcxababxababzxcxcxababxababz+=+++=+=+++=][][][][2222122122212211212122121211122112111111ababcababcababcababc+=+=+=+=)2,1;2,1(21===∑=jiabckkjikij12Matrix Multiplication III• Coefficients as matrix components⎥⎦⎤⎢⎣⎡++++=⎥⎦⎤⎢⎣⎡=2222122121221121221212112112111122211211ababababababababccccC)2,1;2,1(21===∑=jiabckkjikij⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=222112112221121122211211ccccbbbbaaaaCBA313General Matrix Multiplication• For matrix multiplication, C = AB– A has n rows and p columns– B has p rows and m columns– C has n rows and m columns),1;,1(1mjniabcpkkjikij===∑=⎥⎦⎤⎢⎣⎡−=⎥⎦⎤⎢⎣⎡+−+−−+−+=⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡=⎥⎦⎤⎢⎣⎡−−=1210627)1(0)2(2)4(4)6(0)1(2)3(4)1(6)2(0)4(3)6(6)1(0)3(3162143024603ABBA•Example14Coordinate transformations• Recall previous equations⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=21212221121122211211yyxxbbbbaaaayxBA2221212212111122212122121111ybybzybybzxaxayxaxay+=+=+=+=• Define matrices so that y = Ax and z = By15Coordinate transformations II• Define C matrix such that z = Cx⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡=2122211211zzcccczC• From previous slide, y = Ax and z = By• Combine these two to eliminate yz = By = BAxBut, z = Cx so, C = BA• Represent systems of equations 22212122121111xcxczxcxcz+=+=16Matrix Multiplication Exercise• Consider the following matrices• Can you find AB or BA?• We can find AB, because A has three columns and B has three rows• We cannot find BA because B has four columns and A has three rows⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=011410321A⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=131201310220B17Matrix Multiplication Exercise II• What is the size of C = AB?• C = AB has three rows (like A) and four columns (like B)• What is c11?•c11= (1)(0) + (2)(1) + (3)(2) = 8⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=011410321A⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=131201310220B18Matrix Multiplication Exercise III• What are c12, c13, and c14in C = AB?•c11= (1)(0) + (2)(1) + (3)(2) = 8•c12= (1)(2) + (2)(3) + (3)(1) = 11•c13= (1)(-2) + (2)(1) + (3)(-3) = -9•c14= (1)(0) + (2)(0) + (3)(1) = 3⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=011410321A⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=131201310220B419Matrix Multiplication Exercise III• Find c21, c22, c23, and c24in C = AB•c21= (0)(0) + (-1)(1) + (4)(2) = 7•c22= (0)(2) + (-1)(3) + (4)(1) = 1•c23= (0)(-2) + (-1)(1) + (4)(-3) = -13•c24= (0)(0) + (-1)(0) + (4)(1) = 4⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=011410321A⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=131201310220B20Matrix Multiplication Exercise IV• Find c31, c32, c33, and c34in C = AB•c31= (1)(0) + (1)(1) + (0)(2) = 1•c32= (1)(2) + (1)(3) + (0)(1) = 5•c33= (0)(-2) + (1)(1) + (0)(-3) = 1•c34= (1)(0) + (1)(0) + (0)(1) = 0⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−=011410321A⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡−−=131201310220B21Determinants• Looks like a matrix but isn’t a matrix• A square array of numbers with a rule for computing a single value for the array132231331221233211231231133221332211333231232221131211333231232221131211aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaDet−−−++==⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡Example at right shows calculation of determinant for a 3 x 3 matrix A22General Determinant• The value of an n by n determinant is found by taking a sum of n! terms• Each term in the sum has –


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