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Unit18

# Unit18 - Unit 18 Determinants Every square matrix has a...

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Unit 18 Determinants Every square matrix has a number associated with it, called its determi- nant. In this section, we determine how to calculate this number, and also look at some of the properties of the determinant of a (square) matrix. Calculating the determinant of a small matrix is easy. But as the order of the square matrix increases, the calculation of the determinant becomes more complicated. We deﬁne the determinant recursively. That is, for a square matrix of order 1 or 2, we will deﬁne a simple formula for calculating the determinant. However, for a square matrix of order 3 or higher, we deﬁne the determinant of the matrix in terms of the determinants of certain submatrices. Thus, we calculate the determinant of a large matrix by calculating the determinants of various smaller matrices. Deﬁnition 18.1. The Deﬁnition of Determinant, Part 1 ( n 2) Let A =( a ij )bean n × n matrix. Then A has associated with it a number, called the determinant of A and denoted by detA .F o r n = 1 or 2, the value of detA is calculated as follows: If n =1 ,then detA = a 11 . If n =2 detA = a 11 a 22 - a 12 a 21 . Example 1. Find the determinants of the following matrices: (a) A =[ - 4] (b) B = ± ab cd ² (c) C = ± 12 - 35 ² Solution: (a) Here, A is 1 × 1, so detA = a 11 = - 4. That is, for a 1 × 1 matrix, the determinant is the single number in the matrix. 1

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(b) For a 2 × 2 matrix, we use the formula detB = b 11 b 22 - b 12 b 21 .Tha ti s ,w e take the product of the numbers going diagonally down to the right (i.e., on the main diagonal) and then subtract from that the product of the numbers going diagonally down to the left (i.e., on the other diagonal). So assuming that a , b , c and d in matrix B are any scalars, the number detB is given by detB = ad - bc . (c) Again, we have a 2 × 2 matrix, so we do the same calculation as in (b). We get detC = c 11 c 22 - c 12 c 21 = (1)(5) - (2)( - 3) = 5 - ( - 6) = 11. Now we are ready to deﬁne the notation and terminology needed to build up to the deﬁnition of detA for A a square matrix of order n 3. We will need to express certain submatrices of a matrix, as well as the determinant of such a submatrix and also a particular scalar multiple of this number. These are the focus of the next 2 deﬁnitions, after which we will be able to complete our deﬁnition of determinant.
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Unit18 - Unit 18 Determinants Every square matrix has a...

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