By Donald L. Kreider, Robert G. Kuller, Donald R. Ostberg, Fred W. Perkins, Lynn H. LOoomis
An advent to Linear research
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Additional info for An Introduction to Linear Analysis (Addison-Wesley Series in Mathematics)
Verify Property 5 for λ = — 3 and 1 0' 1 3 1 1 Α= 5. Verify Property 8 for A= 6 1 1 2 and B= 6. Without expanding, show that 3r a r x 2a X Ab 6s 2y = - 1 2 b s y 2c -3r — z c t z 7. 4 PIVOTAL CONDENSATION By making use of the properties of determinants given in the previous section, it is often possible to transform a specific determinant into another equivalent determinant which is easier to evaluate. One such procedure is the method of pivotal condensation. This method involves operating on a determinant so that one row or one column is transformed into a new row or column containing at most one nonzero element.
4. Find |A| if A= 1 4 1 2 A = 11 2 4 3 0 Find |A| if 4 2 0 1 1 3 1 0 -2 1 1 0 9 1 1 0 - 1 1 0 2 2 1 0 1 2 0" 0 0 0 0 Can you generalize this result to any arbitrary matrix containing a zero column or zero row ? 3 PROPERTIES OF DETERMINANTS In this section, we give some useful properties of determinants. For the sake of expediency, we present most of these properties in terms of row operations; they are, however, equally valid for the analogous column operations. Similarly, we only prove each property for determinants of order three keeping in mind that, in general, these proofs may be extended, in a straightforward manner, to cover determinants of higher order.
Thus, B _ 1 A _ 1 is also an inverse for AB, and, by uniqueness of the inverse, B _ 1 A _ 1 = (AB)"1. Property 3 (A,A 2 · · · A,)" 1 = KlK-i '" A ^ A f 1 . Proof This is an extension of Property 2 and, as such, is proved in a similar manner. Caution: Note that Property 3 states that the inverse of a product is not the product of the inverses but rather the product of the inverses commuted. 3 50 Property 4 The inverse (A)1 =(A1)/ Proof (A')~* denotes the inverse of A'. However, using the property of the transpose that (AB)' = ΒΆ', we have that (Α'ΧΑ-Υ^Α-'Α)'^'^.
An Introduction to Linear Analysis (Addison-Wesley Series in Mathematics) by Donald L. Kreider, Robert G. Kuller, Donald R. Ostberg, Fred W. Perkins, Lynn H. LOoomis