Unit15 - Unit 15 Matrix Operations Recall that is said to...

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Unit 15 Matrix Operations Recall that: A = ( a ij ) = a 11 a 12 a 13 · · · a 1 n a 21 a 22 a 23 · · · a 2 n a 31 a 32 a 33 · · · a 3 n . . . . . . . . . . . . . . . a m 1 a m 2 a m 3 · · · a mn is said to be an m × n matrix, where m × n is called the dimension of A . That is, a matrix whose dimension is m × n has m rows and n columns. and also that: the ( i, j )-entry of A , denoted by a ij , is the entry (i.e., number) in row i and column j of matrix A . That is, for the ( i, j )-entry, i is the row number, and j is the column number, where the rows are numbered 1 to m , from the top down, and the columns are numbered 1 to n , from left to right. Definition 15.1. Let A be any m × n matrix, and let k and t be integers with 1 k n and 1 t m . We use a k to denote the k th column of matrix A , and a t to denote the t th row of matrix A . For instance, for A as shown above, we have: a 2 = a 12 a 22 a 32 . . . a m 2 , a 3 = bracketleftbig a 31 a 32 a 33 · · · a 3 n bracketrightbig Notice: In a k , all entries have the same second subscript k (because all are in the same column), while in a t , all entries have the same first subscript t (because all are in the same row). 1
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Definition 15.2. Two matrices, A and B , are equal , written A = B , if they have the same dimension and their corresponding entries are equal. That is, A and B are equal if both are m × n matrices (for the same values of m and n ) and if a ij = b ij for each i and j . Example 1 . State whether matrices A and B are equal. (a) A = bracketleftbigg 1 - 2 3 4 0 6 bracketrightbigg B = bracketleftbigg 1 - 2 3 4 0 6 bracketrightbigg (b) A = bracketleftbigg 1 0 3 5 1 - 2 bracketrightbigg B = bracketleftbigg 1 0 3 5 1 2 bracketrightbigg (c) A = bracketleftbigg 1 2 3 4 5 6 bracketrightbigg B = 1 4 2 5 3 6 (d) A = bracketleftbigg 1 0 0 1 bracketrightbigg B = 1 0 0 1 0 0 Solution: (a) A = bracketleftbigg 1 - 2 3 4 0 6 bracketrightbigg = B = bracketleftbigg 1 - 2 3 4 0 6 bracketrightbigg Since A and B both have dimension 2 × 3 and a ij = b ij for each pair ( i, j ), they are equal matrices. (b) A = bracketleftbigg 1 0 3 5 1 - 2 bracketrightbigg negationslash = B = bracketleftbigg 1 0 3 5 1 2 bracketrightbigg Although A and B are both 2 × 3 matrices, with many of their entries iden- tical, there is a combination ij for which a ij negationslash = b ij ( i.e. a 23 = - 2 whereas b 23 = 2). Therefore, A and B are not equal matrices. (c) A = bracketleftbigg 1 2 3 4 5 6 bracketrightbigg negationslash = B = 1 4 2 5 3 6 Here, A has dimension 2 × 3, whereas B has dimension 3 × 2, so they cannot be equal matrices, no matter how similar their entries may be. 2
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(d) A = bracketleftbigg 1 0 0 1 bracketrightbigg negationslash = B = 1 0 0 1 0 0 Again, A and B do not have the same dimension ( A is 2 × 2 while B is 3 × 2), so they are not equal.
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