This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Solutions to Homework #9, MAT1341A Question # 1: (3+3=6points) Compute the determinants of the following matrices. (a) A = 3 1 2 3 5 2 1 4 Solution: We have several options. Method 1: By direct cofactor expansion, on row 1: det( A ) = 3 det 3 5 1 4 ( 1) det 2 5 2 4 = 3(7) + (18) = 39 Method 2: First do some row reduction, and keep track of steps: A 3 2 R 2 + R 1 ∼ R 2 + R 3  11 / 2 15 / 2 2 3 5 4 9 R 1 ↔ R 2 ∼ 2 3 5 11 / 2 15 / 2 4 9  2 / 11 R 2 ∼ 2 3 5 1 15 / 11 4 9  4 R 2 + R 3 ∼ 2 3 5 1 15 / 11 39 / 11 = R which is triangular and has determinant equal to the product of the diagonal entries, so det( R ) = 78 / 11. Now we have det( R ) = ( 1)( 2 / 11) det( A ) because of the row interchange and the scaling, and so det( A ) = 39 Method 3: Use a clever combination of techniques: det( A ) = det( A T ) = det 3 2 2 1 3 1 5 4 = det 11 1 1 3 1 5 4 = ( 1) det 11 1 5 4 = 44 5 = 39 (we did a row reduction step: 3R2+R1, which doesn’t change the determinant). (b) B =  3 1 5 1 1 3 2 3 4 1 2 3 4 Solution: We could do a direct cofactor expansion along any row or column; and certainly we would try to use a column with the most zeros; in this case, 1. Say we choose column 4; then recalling the signs on the cofactors are given by the formula ( 1) i + j we have det( A ) = det 1 3 3 4 2 3 4 + 2 det  2 1 5 3 4 2 3 4  det  3 1 5 1 3 3 4 and then you keep expanding. However, this involves a lot of calculation and is therefore quite errorprone; you would be much better off doing a few simple row reduction steps to create more zeros. For instance, you could do 2 R 1 + R 2 and R 1 + R 3; and that would zero out all but one entry in column 4, so instead of three 3 ×...
View
Full
Document
This note was uploaded on 09/28/2011 for the course MATH 1341 taught by Professor Daigle during the Winter '10 term at University of Ottawa.
 Winter '10
 DAIGLE
 Determinant, Matrices

Click to edit the document details