hw7sol - Winter 2011 Math 67 Linear Algebra Homework 7...

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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 i 2- i 3 = 2- 2 i (1) 1 1 1- 1 =- 2 (2) 2 0 1 0 1 0 1 0 2 = 3 (3) 1 0 0 0 5 2 0 0 8 6 3 0 0 9 7 4 = 24 (4) 1 4 1 3 2 3 5 0 4 1 0 0 2 0 0 0 = 30 (5) Problem B. (this is Artin 1.3.8) Let A be an n-by- n matrix. What is det(- A )? (You do not need to hand this in.) Solution. The answer is (- 1) n A . The reason is that each row of- A is a row of A multiplied by- 1. Factoring out n copies of- 1 gives (- 1) n A . Problem C. Let P and Q be matrices of size n-by- n and m-by- m . Form a new matrix of the form P Q This matrix has dimensions ( n + m )-by-( n + m ). Prove that det P Q = (det P )(det Q ) using the definition of the determinant. Do any example where n = m = 2, but do not hand this part in. 1 2 Solution. Let the i,j entry of P by a i,j . If P i,j denotes P with row i and column j crossed out. Then det R = a 1 , 1 det P 1 , 1 Q- a 1 , 2 det P 1 , 2 Q + a 1 , 3 det...
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hw7sol - Winter 2011 Math 67 Linear Algebra Homework 7...

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