b Reduced row echelon form 118 1 1 1 1 1 3 2 1 1 1 2 4 1 1 1 1 1 5 1 1 1 4

B reduced row echelon form 118 1 1 1 1 1 3 2 1 1 1 2

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(b) Reduced row echelon form. 118
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1 1 1 1 1 3 2 1 1 1 2 4 1 - 1 - 1 1 1 5 1 0 0 1 1 4 . Solution: (a) 1 1 1 1 1 3 2 1 1 1 2 4 1 - 1 - 1 1 1 5 1 0 0 1 1 4 1 1 1 1 1 3 0 - 1 - 1 - 1 0 - 2 0 - 2 - 2 0 0 2 0 - 1 - 1 0 0 1 R 2 - 2 R 1 R 3 - R 1 R 4 - R 1 1 1 1 1 1 3 0 1 1 1 0 2 0 - 2 - 2 0 0 2 0 - 1 - 1 0 0 1 - R 2 1 1 1 1 1 3 0 1 1 1 0 2 0 0 0 2 0 6 0 0 0 1 0 3 R 3 + 2 R 2 R 4 + R 2 1 1 1 1 1 3 0 1 1 1 0 2 0 0 0 1 0 3 0 0 0 1 0 3 1 2 × R 3 1 1 1 1 1 3 0 1 1 1 0 2 0 0 0 1 0 3 0 0 0 0 0 0 R 4 - R 3 119
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This matrix is now in a row echelon form. (b) Now to obtain a reduced row echelon form, we continue from the last step in ( a ) to ensure that all columns with leading one have zeros elsewhere as follows: 1 1 1 0 1 0 0 1 1 0 0 - 1 0 0 0 1 0 3 0 0 0 0 0 0 R 1 - R 3 R 2 - R 3 1 0 0 0 1 0 0 1 1 0 0 - 1 0 0 0 1 0 3 0 0 0 0 0 0 R 1 - R 2 Tutorials 5.2.6. Reduce the following matrix and leave your answer in (a) Row echelon form. (b) Reduced row echelon form. 1. 1 2 - 3 - 2 4 1 2 5 - 8 - 1 6 4 1 4 7 5 2 8 . 2. 1 2 1 2 1 2 2 4 3 5 5 7 3 6 4 9 10 11 1 2 4 3 6 9 . 5.3 Determinant Given any square matrix A , there is a special scalar called the determinant of A denoted by det( A ) or we make use of vertical bars | A | . This number is also important in determining whether a matrix is invertible or not, which will be studied in the next section. We will show how to evaluate determinant of square matrices of different orders. 120
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(a): 1 × 1 matrix: If A = ( a ), then we simply say det A = a . Example 5.3.1. A = (2) gives det A = 2 . (b): 2 × 2 matrix: If A = a b c d ; The determinant is defined by: det A = a b c d = ad - bc Example 5.3.2. If A = 5 - 1 6 - 3 ,then det A = 5 - 1 6 - 3 = (5)( - 3) - 6( - 1) = - 15 + 6 = - 9 . Definition 5.3.3. The ( i, j ) -th minor of a matrix A , denoted by m ij ( A ) , is the determinant obtained by deleting row i and column j of A . e.g. If A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 , then m 32 ( A ) = a 11 a 13 a 21 a 23 . Definition 5.3.4. The ( i, j ) -th cofactor of a matrix A , denoted by c ij ( A ) , is the signed minor given by c ij ( A ) = ( - 1) i + j m ij ( A ) of A . e.g. If A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 , then c 23 ( A ) = ( - 1) 2+3 a 11 a 12 a 31 a 32 = - a 11 a 12 a 31 a 32 . 121
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(c): 3 × 3 matrix: (i) Laplace expansion: Given A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ; det A = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 ( - 1) 1+1 m 11 + a 12 ( - 1) 1+2 m 12 + a 13 ( - 1) 1+3 m 13 = a 11 ( - 1) 2 a 22 a 23 a 32 a 33 + a 12 ( - 1) 3 a 21 a 23 a 31 a 33 + a 13 ( - 1) 4 a 21 a 22 a 31 a 32 . This is Laplace expansion along row one. This expansion can be done along any other row or column without affecting the value of the determi- nant. Example 5.3.5. A= - 2 3 - 1 4 - 7 1 1 2 - 5 det A = - 2 3 - 1 4 - 7 1 1 2 - 5 Expand along row 1 = ( - 2)( - 1) 1+1 - 7 1 2 - 5 + 3( - 1) 1+2 4 1 1 - 5 + ( - 1)( - 1) 1+3 4 - 7 1 2 = ( - 2)(35 - 2) - 3( - 20 - 1) - 1(8 + 7) = - 66 + 63 - 15 = - 18 . (ii) Sarrus scheme: 122
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Another useful method which can be used to evaluate the determinant of a 3 × 3 matrix is what we call “ Sarrus scheme ” or also commonly referred to as butterfly method ”. To find the det( A ) using this method we follow the following steps: (i) Rewrite the 1 st and 2 nd columns on the right (as Columns 4 and 5).
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