1.1 1.2 1.3 1.4 1.5 1.6 Matrices Operations of matrices Types of matrices Properties of matrices Determinants Inverse of a 3>
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1.1 Matrices 237 1 131 476 Both A and B are examples of matrix. A matrix is a rectangular array of numbers enclosed by a pair of bracket. Why matrix?
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1.1 Matrices Consider the following set of equations: It is easy to show that x = 3 and 3x-y=5e x+y-2z=7, 2x-y-4z = 2, How about solving —5x + 4 Y' 10 z = 1, 3x—y—6z = 5. Matrices can help... 4
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1.1 Matrices In the matrix 21 ml 12 22 In 2n mn •numbers q. are called elements. First subscript indicates the row; second subscript indicates the column. The matrix consists of mn elements •It is called "the m x n matrix A = or simply "the matrix A " if number of rows and columns are understood.
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1.1 Matrices Square matrices •When m = n, i.e., 21 12 22 In 2n nn •A is called a "square matrix of order n" or n-square matrix" •elements a a a33,..., ann called diagonal elements. an + is called the trace of A.
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1.1 Matrices Equal matrices •Two matrices A = [at..] and B = are said to be equal (A = B) iff each element of A is equal to the corresponding element of B, i.e., b for 1 < i < m, 1 < j < n. •iffpronouns "if and only if" if A = B, it implies bi for 1 < i < m, 1 n; if bi for 1 S i Sm, I s j < n, it implies A = B.
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1.1 Matrices Equal matrices Example: A and B Given that A = B, find a, b, cand d. if A = B, then a = 1, 0, c = -4 and d=2
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1.1 Matrices Zero matrices •Every element of a matrix is zero, it is called a zero matrix, i.e., 0 0 0 O O 0 O O 0
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1.2 Operations of matrices Sums of matrices •If A = and B = are m x n matrices, then A + B is defined as a matrix C = A + B, where [Cid, c = q.. + bi. for 1 1 < j < n. 123 230 Example: if A and B Evaluate A + B and A-B 1-2 2-3 3+0 3-0 4—5 3 —l 1 5 3 -1 3 9 3 -1 10
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1.2 Operations of matrices Sums of matrices •Two matrices of the same order are said to be conformable for addition or subtraction. •Two matrices of different orders cannot be added or subtracted, e.g., 131 are NOT conformable for addition or subtraction.
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1.2 Operations of matrices Scalar multiplication •Let X be any scalar and A = [at..] is an m x n matrix. Then IA = [laze.] for 1 m, 1 < j < n, i.e., each element in A is multiplied by 1. 431 Example: A . Evaluate 3A. •In particular, 1 e A — ; = [-4.1. It's called the negative of A. Note: A —A = 0 is a zero matrix
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1.2 Operations of matrices Properties Matrices A, B and C are conformable, (commutative law) .A + (B +0 = (A + B) +C (associative law) + B) = + where is a scalar (distributive law) Can you prove them?
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1.2 Operations of matrices Properties Example: Prove + B) = IA + 1B, Let C=A+B, so c Consider = (a. Since lc = + B), + hi.) = Ia.. + we have, so + B) = +
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1.2 Operations of matrices Matrix multiplication •If A = is a m x p matrix and B = [bid is a p x n matrix, then AB is defined as a m x n matrix C = AB, where C= with for 1 i m, 1 < j n. C. Example: A Evaluate 1 1 0 2 1 3 2 3 5 c 21 3 —Ox( and c = AB. —1) + X 2 + X 5 22 15
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1.2 Operations of matrices Matrix multiplication 123 Example: A B = 2 3 , Evaluate C = AB. C - AB 1 0 = 18 22 1 o 2 1 3 4 12 = OX -k 4X5 21 22 18 8 22 3
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Operations of matrices Matrix multiplication •In particular, A is a 1 x m matrix and B is am x 1 matrix, i.e., A — [all an then C = AB is a scalar. c m k=l 11 21 ml — al 11911 + al 2b 21 + + a b 1m ml 17
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Operations of matrices Matrix multiplication •BUT BA is am x m matrix! 11 21 ml an 11 11 21 11 ml Il 11 12 21 12 ml 12 •So AB in general ! Il 1m 21 1m ml 1m 18
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1.2 Operations of matrices Properties Matrices A, B and C are conformable, '(A + = AC+BC •A(BC) = (AB) C •AB in general •AB = 0 NOT necessarily imply A = 0 or B - X •AB = AC NOT necessarily imply B = C
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1.2 Operations of matrices Properties Example: Prove + C) = AB + AC where A, B and C are n-square matrices Let X = B + C, so Xi. = Let Y = AX, then Yij = = + Ck.) ik kj so Y = AB + AC; therefore, VA(B + C) - 20
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Types of matrices •Identity matrix •The inverse of a matrix •The transpose of a matrix •Symmetric matrix •Orthogonal matrix
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1.3 Types of matrices Identity matrix •A square matrix whose [email protected] = 0, for i > j is called upper triangular, i.e., q 12 0 a 22 O O •A square matrix whose elements a = 0, for i < j is called lower triangular, i.e., a 0 21 22 In 2n nn nn 22
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1.3 Types of matrices Identity matrix •Both upper and lower triangular, = 0, for O al 0 an O O O a nn is called a diagonal matrix, simply 23
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1.3 Types of matrices Identity matrix •In particular, all - ann= 1, the — an matrix is called identity matrix. •Properties: Al = IA = A Examples of identity matrices: 100 and 0 1 0 001 24
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1.3 Types of matrices Special square matrix •AB n BA in general. However, if two square matrices A and B such that AB = BA, then A and B are said to be commute. Can you suggest two matrices that must commute with a square matrix A? 'X!U4DW Å4!4uep! V :SuV •If A and B such that AB = -BA, then A and B are said to be anti-commute, 25
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1.3 Types of matrices The inverse of a matrix •If matrices A and B such that AB = BA = I, then B is called the inverse of A (symbol: A-I); and A is called the inverse of B (symbol: B-l). 123 6 Example: A = 1 33 B Show B is the the inverse of matrix A. 1 Ans: Note that o 0 I o Can you show the details? 0 O 1 26
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1.3 Types of matrices The transpose of a matrix •The matrix obtained by interchanging the rows and columns of a matrix A is called the transpose of A (write AT). Example: A The transpose of A is AT = 2 5 •For a matrix A = Lao.] , its transpose AT = [bij], where b 27
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Types of matrices Symmetric matrix •A matrix A such that AT = A is called symmetric i.e., aft = for all i and j. •A + AT must be symmetric. Why? 123 Example: A = 2 4 5 is symmetric. 3 •A matrix A such that AT = -A is called skew- symmetric, i.e., ajl = -q. for all i and j. •A -AT must be skew-symmetric. Why? 28
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1.3 Types of matrices Orthogonal matrix •A matrix A is called orthogonal if AAT = ATA Example: prove that A = 1/6 -2/Uö o is orthogonal. Since, -2/6 . Hence, = l. Can you show the detai Is? We'll see that orthogonal matrix represents a rotation in fact! 29
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1.4 Properties of matrix •(AB)-I = B-lA-l = A and = •(AB)T = WAT 30
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1.4 Properties of matrix Example: Prove (AB)-I = B-lA-l and (AB) (B-IA-I) = B-l)A-l = 1 Since (B-IA-I) (AB) = B-l(A-l = 1 Therefore, is the inverse of matrix B-lA-l AB
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1.5 Determinants Determinant of order 2 Consider a 2 x 2 matrix: A a 21 an a 22 •Determinant of A, denoted I Al, is a number and can be evaluated by an an 11 22 an 12 21 32
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1.5 Determinants Determinant of order 2 •easy to remember (for order 2 only).. a 2 11 22 a 1 12 21 12 Example: Evaluate the determinant: 12 -2 33
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1.5 Determinants The following properties are true for determinants of any order. 1. 2. 3. If every element of a row (column) is zero, 12 , then IAI = 0. e.g., determinant of a matrix = that of its transpose I ABI = 34
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1.5 Determinants Example: Show that the determinant of any orthogonal matrix is either +1 or —1. For any orthogonal matrix, _AAT = 1. Since = = 1 and IA T I = SO or 35
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1.5 Determinants For any 2x2 matrix A 11 21 1 an Its inverse can be written as A Example: Find the inverse of A The determinant of A is -2 01 -1 Hence, the inverse of A is A How to find an inverse for a 3x3 matrix? 36
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1.5 Determinants of order 3 123 Consider an example: A = 4 5 6 Its determinant can be obtained by: 123 12 12 IAI=4 5 —6 +9 You are encouraged to find the determinant by using other rows or columns 37
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Inverse of a 3>
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1.6 Inverse of a 3>
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Inverse matrix of A Inverse of a 3>
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