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04-02-03 - Lecture 23 Andrei Antonenko April 2 2003 1...

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Lecture 23 Andrei Antonenko April 2, 2003 1 Properties of determinants-2 Now we’ll give a first motivation of the determinant. Theorem 1.1 (Criteria of invertibility). A square matrix A is invertible if and only if det A 6 = 0 . Proof. Let’s use elementary row operations to transform a matrix A to its triangular (row- echelon) form. Let’s note, that if the determinant was not equal to 0, then it will not be equal to 0 after elementary row operations, and if it was equal to 0, it will be equal to 0 after elementary row operations. The matrix A is invertible, if it’s REF doesn’t have a row of zeros, i.e. the determinant of its REF is not equal to 0. So, the determinant of the initial matrix is not equal to 0. Moreover, A is not invertible if we have a row of zeros in its REF, so the determinant of REF equals to 0, and so, the determinant of the initial matrix A is equal to 0. Example 1.2. We computed the following determinant: fl fl fl fl fl fl fl 1 2 3 4 5 6 7 8 9 fl fl fl fl fl fl fl = 0 . So, this matrix is not invertible. This theorem gives us a criteria, when the matrix is invertible. But if we know that the determinant is not equal to 0, and so the matrix is invertible, the theorem doesn’t give us a method of computing the inverse. Now let’s continue with properties of determinants. Theorem 1.3 (The determinant of the transpose). det A > = det A . Example 1.4. Since fl fl fl fl fl fl fl 1 2 3 4 5 6 7 8 9 fl fl fl fl fl fl fl = 0 , we have that fl fl fl fl fl fl fl 1 4 7 2 5 8 3 6 9 fl fl fl fl fl fl fl = 0 . 1
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Proof. The determinant of A > is equal to the sum of all possible products of matrix elements, taken 1 from each column and 1 from each row, as well as the determinant of A . So we have to check that the products are included with the same signs. To figure out the sign before a 1 k 1 a 2 k 2 . . . a 3 k 3 in the expression for the determinant of A > , we have to reorder multiplicands by the second subscript. So, we can interchange multiplicands, and transpositions will occur simultaneously in the first subscripts and in the second subscripts.
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