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2003 - 2004 Possible MT1 Questions

# 2003 - 2004 Possible MT1 Questions - MATH 250 BASIC LINEAR...

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Unformatted text preview: MATH 250 - BASIC LINEAR ALGEBRA SUPPLEMENTARY PROBLEMS 1 These are the questions Lhat were suggested by [he washers while preparing th: ﬁrsl midterm examination in Malh 260 - 53ch Linear Algebra to take place on 5th April 2004 . 1. Let A be a square matrix that satisﬁes .43 + 7.4 + I : 0. Prove that detA ;£ 0. 2. “det(AB} = that}! defB hnlds for any square matrices A and B." True or false ? 3. “det(.~’l + B) = delA + detB holds for any Square matrices A and B.” True or false '3 4. Consider the matrix [I u U U 0 b [I c U U A= D :1 I] r2 0 n a f o y El El 0 h. 0 Prove that. A is singuiau' for may values of (1,15, c, die,f.g e [K 5. 1:, it possible for nan-singular n x n—matricrs 31.3, P. Q to satisfy ABA'U‘S" : 3PQF"Q“ ? B. Let H : I — 2PPT‘ where P is an n x l maLrix such that PTP : [ 1 ] A Prove that H7 : H" = H 7. Compute the inverses, if any, of the matrices 1 2 3 3 A: 4 Eu 6 , B 6 10 7 8 9 I. Let n 2 2 and A = [why-4:5,. where a” = 1 for all 1 S 1}} S n. Prove that I + A is a non-singular matrix with il up..- acute -1 f _ (1+ A) _ 1 n _ 1A 9. For which values uf (1.0 e B does the system a: + + bz = 2 am + ay + 42 = 4 ﬂy + 2z = b have (i) a unique solution (ii) no solution (iii) solutions depending upon exactly one arbitrary parameter 7 10. Give an example of a. square matrix A with the property that A3 = 0 and A? # D. For any such A prove that I + A is a. non-singular matrix by ﬁnding (1' + A)"1 . 11. Suppose that the square matrix A has the property A2 : A. Prove that either detA=UDrA=I. 12. Prove that a —b —c —d b a d —c det c —d a b = (I12+b2+c2+d2)2 ...
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2003 - 2004 Possible MT1 Questions - MATH 250 BASIC LINEAR...

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