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Topic3 - Matrix A matrix is a rectangular array of elements...

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Matrix A matrix is a rectangular array of elements arranged in rows and columns Dimension of a matrix is r x c r = c  square matrix r = 1  (row) vector c = 1  column vector 1 4 2 5 3 6 = A
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Matrix (cont) elements are either numbers of symbols each element is indicated by aij rows are indicated by an i subscript columns are indicated by a j subscript each element is designated by the row and column that it is in 11 12 21 22 ij 31 32 1 4 a a 2 5 a a a ,i 1 3,j 1 2 3 6 a a = = = = - = - A
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Matrix Equality two matrices are said to be equal if they have the same dimensions and all of the corresponding elements are equal
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Equality: Example Are the following matrices equal? a) b) 1 4 1 2 3 2 5 4 5 6 3 6 = = A B 1 4 1 4 2 5 3 5 3 6 3 6 = = A B
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Matrix Operations - Transpose A transposition is performed by switching the rows and columns, indicated by a ‘ A’ = B if aij = bji Note: if A is r x c then B will be c x r The transpose of a column vector is a row vector. 1 4 1 2 3 2 5 ' 4 5 6 3 6 = = A A
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Matrix Operations – Addition/Subtraction Two add or subtract two matrices, they must have the same dimensions The addition or subtraction is done on an element by element bases 1 4 10 40 2 5 , 20 50 3 6 30 60 = = A B 1 10 4 40 11 44 2 20 5 50 22 55 3 30 6 60 33 66 + + = + + = + + A +B
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Matrix Operations - Multiplication by a scalar (number) every element of the matrix is multiplied by that number In addition, a matrix can be factored if A = [aij], k is a number then k A = A j = 2 4 2k 4k ,k 3 5 3k 5k = = A A 2 4 2 4 1 k k k 3 5 3 5 k k =
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Matrix Operations - Multiplication by another matrix C = A B columns of A must equal rows of B Resulting matrix has dimension rows of A x columns of B ij ik kj k c a b = 1 2 4 2 1 4 8 2 8 2 3 14 13 4 1 1 2 1 1 8 2 2 2 3 11 7 2 3 + + + + = = =
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