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Unformatted text preview: Winter 2011 • • Math 67 Linear Algebra Homework 7 Problem A. (this is Artin 1.3.1) Compute the following determinants:
(1) 1
i
2−i 3 (2) 11
1 −1 (3) 201
010
102 (4) 1
5
8
0 0
2
6
9 0
0
3
7 0
0
0
4 (5) 1
2
4
2 4
3
1
0 1
5
0
0 3
0
0
0 Problem B. (this is Artin 1.3.8) Let A be an nbyn matrix. What is
det(−A)? (You do not need to hand this in.) Problem C. Let P and Q be matrices of size nbyn and mbym. Form a
new matrix of the form
P0
0Q
This matrix has dimensions (n + m)by(n + m). Prove that
det P0
= (det P )(det Q)
0Q using the deﬁnition of the determinant. Do any example where n = m = 2,
but do not hand this part in. Problem D. Compute the eigenvalues of the matrix in A(5), above. (Use a
calculator to compute the zeros of the polynomial det(A − λI ). There is one
online at
http://www.convertalot.com/quartic_root_calculator.html)
1 2 Problem F. Compute the inverse of 111
2 3 4 4 9 16
using Cramer’s rule. Problem G. Suppose that A is an upper triangular matrix. Prove that A−1
is also upper triangular using Cramer’s rule. ...
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This note was uploaded on 11/13/2011 for the course MATH 67 taught by Professor Schilling during the Winter '07 term at UC Davis.
 Winter '07
 Schilling
 Linear Algebra, Algebra, Determinant

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