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la03-d - LINEAR ALGEBRA W W L CHEN c W W L Chen 1982 2005...

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LINEAR ALGEBRA W W L CHEN c W W L Chen, 1982, 2005. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners. Chapter 3 DETERMINANTS 3.1. Introduction In the last chapter, we have related the question of the invertibility of a square matrix to a question of solutions of systems of linear equations. In some sense, this is unsatisfactory, since it is not simple to find an answer to either of these questions without a lot of work. In this chapter, we shall relate these two questions to the question of the determinant of the matrix in question. As we shall see later, the task is reduced to checking whether this determinant is zero or non-zero. So what is the determinant? Let us start with 1 × 1 matrices, of the form A = ( a ) . Note here that I 1 = (1). If a = 0, then clearly the matrix A is invertible, with inverse matrix A 1 = ( a 1 ) . On the other hand, if a = 0, then clearly no matrix B can satisfy AB = BA = I 1 , so that the matrix A is not invertible. We therefore conclude that the value a is a good “determinant” to determine whether the 1 × 1 matrix A is invertible, since the matrix A is invertible if and only if a = 0. Let us then agree on the following definition. Definition. Suppose that A = ( a ) is a 1 × 1 matrix. We write det( A ) = a, and call this the determinant of the matrix A . Chapter 3 : Determinants page 1 of 23
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Linear Algebra c W W L Chen, 1982, 2005 Next, let us turn to 2 × 2 matrices, of the form A = a b c d . We shall use elementary row operations to find out when the matrix A is invertible. So we consider the array (1) ( A | I 2 ) = a b 1 0 c d 0 1 , and try to use elementary row operations to reduce the left hand half of the array to I 2 . Suppose first of all that a = c = 0. Then the array becomes 0 b 1 0 0 d 0 1 , and so it is impossible to reduce the left hand half of the array by elementary row operations to the matrix I 2 . Consider next the case a = 0. Multiplying row 2 of the array (1) by a , we obtain a b 1 0 ac ad 0 a . Adding c times row 1 to row 2, we obtain (2) a b 1 0 0 ad bc c a . If D = ad bc = 0, then this becomes a b 1 0 0 0 c a , and so it is impossible to reduce the left hand half of the array by elementary row operations to the matrix I 2 . On the other hand, if D = ad bc = 0, then the array (2) can be reduced by elementary row operations to 1 0 d/D b/D 0 1 c/D a/D , so that A 1 = 1 ad bc d b c a .
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